There are two things: statistical significance and clinical significance.  You have already achieved statistical significance, so having a larger sample size will not change anything (expect making your p smaller).  What you need to show now is that your finding had a clinical significance.  Comparing the mean to SD gives one way to measure the effect size (and yours seems pretty good, on the face of it).  In addition, each field has a favourite measure of effect size, which helps the reader appreciate whether your particular finding is going to change the world for the better.  My personal preference in healthcare is the number needed to treat (or number needed to harm). 

There are two things: statistical significance and clinical significance.  You have already achieved statistical significance, so having a larger sample size will not change anything (expect making your p smaller).  What you need to show now is that your finding had a clinical significance.  Comparing the mean to SD gives one way to measure the effect size (and yours seems pretty good, on the face of it).  In addition, each field has a favourite measure of effect size, which helps the reader appreciate whether your particular finding is going to change the world for the better.  My personal preference in healthcare is the number needed to treat (or number needed to harm). 

Altghoug Alexei really hit the most important point (statistical significance as such is NOT equal to - not even related to! - clinical relevance), I want to add some background:

Such questions are frequent, and the source of this kind of confusion is a logically flawed application (and teaching!) of "null hypothesis significanse tests".

This is not one method, this is a conglomerate of two completely incompatible attempts to tackle uncertainty and to draw conclusions from some given (noisy) data.

One way is Fisher's significance testing, where the question is whether or not further research with the given limitations of money and time (and samples) will be feasable. For this purpose a kind of a "standardized signal-to-noise ratio" is calculated: the p-value or "statistical significance". This value indicates our expectation to get results from this kind of experiment that are at least as "extreme" under a particular given hypothesis. This hypothesis (the "null hypothesis") represents a state of the world for which a further research would be of no use at all. If the expectation of the observed data (or more "extreme" data) under this hypothesis is really low, one may argue that a further research is in fact feasable. There is no particular cut-off for this conclusion. Which p-value is small enough to justify further research depends on the whole context.

This way has some logical flaws. One problem is that it inherently assumes, in principle, that the probability of a hypothesis given the data, is the same as the probability of the data given some hypothesis. This is simply wrong. The other problem is that really only the practically completely uninteresting null hypothesis is tested and that there is no relevant alternative stated. Rejecting the null hypothesis thus provides simply no information at all about the relevance or reasonability of a decision or conclusion. The same applies in cases where the null hypothesis can not be rejected. It remains completely unclear if this is a smart way to go. The sheet anchor for this way of drawing conclusions is going back to ask: is the estimated effect relevant? - but this could/should have been done in the first place, it depends on the context, is a judgement-call with neccessarily subjective flavour, and the whole "statistical significance" part is actually not really required.

The other way is a purely decision-theoretic approach trying to overcome the logical problems of Fisher's approach of significance testing. Neyman's hypothesis testing is based on the logical impossibility to make any conclusion about a hypothesis given the data without having to make further, often neccessarily subjective prior assumptions about the probabilities of hypotheses. This lead to a strategy of an inductive behaviour. This strategy requires to state a relevant alternative hypothesis and to quantify expected losses with wrong conclusions (of both types: falsely rejecting a correct null hypotheses as well as falsely rejecting a correct alternative hypothesis). The subjective part here is to state acceptable rates of wrong decisions, and such statements can only be reasonable when the expextes losses and benefits are somehow quantified and valued (again in the whole context of the research). Setting arbitrary "commonly used" acceptable rates is clearly unreasonable and can be much more harmful than helpful. Note that the specification of the acceptable rate of fals rejections of the (relevant!) alternative hypothesis requires to adjust the sample size accordingly, so the experiment can be designed and conducted only after having planned all this. Having specified these rates and conducted the planned experiment, however, opens the way to follow some strict rules to get the decision. The possible decisions are: either "behave as if there is no relevant effect" or "behave as if there is a relevant effect". The decision is NOT about the truth or falsity about a hypothesis. It is only about our behavior, that is, about the costs and benefits we expect by doing somthing (based on the data we know), and by controlling the expected maximum losses that we think are possible. The null hypothesis is required only for some internal probability-calculation stuff. The whole concept, however, focuses on the alternative rather than on the null hypothesis.

Now text books and lectures often present a mixture of these two concepts that makes no sense in any way. Experiments are condeucted without specifying a relevant effect (i.e. the alternative hypothesis) and without control of false rejections of the alternative hypothesis. But a rate of acceptable false rejections of the null hypothesis is stated, but not based on some sensible arguments and in a context-specific manner. It is just set to some "commonly used value" (premajorily 0.05), and then the null hypothesis is rejected (and the result is called "significant") whenever the calculated p-values is below this cut-off. If the p-value is larger, the result is deemed "non-significant" and any further conclusion is prohibited.

Now, after all this lengthly text coming back to your question:

"Is there another way to prove that these results are significant despite the high SD?"

There are two problems: "proof" and "significance"

Proofs are a logical construct. They belong in the domain of logic/mathemetics. Here we have data and so we are in the domain of empirical science. There is nothing like a proof for whatsoever in empirical sciences. So: do not seek for proofs. Instead ask for things like plausibility, repeatability, expectation, precision - and relevance.

Significance is somthing that requires a judgement call. Either at the time of planning the whole experiment (Neyman) or at the time of looking at the p-value (Fisher). In Neyman's approach "significance" is finally judged by whether or not a test statistic falls into a "rejection region" (what followes from the judgement call at the state of planning). This is a binary (dichotomous) thing and defines the way you should behave. In Fisher's approach "significance" is the "tinyness of the p-value" what is a continuous, gradual thing. It has no particular objective meaning per se, needs to be judged. Simply comparing it to some arbitrary cut-off is non-sensical (and still if done otherwise the whole method remains flawed unless the effect size is not considered as well).

Dear Wilhelm and Chadyuk,

 Thanks so much for your illuminating answers. Wilhelm, I really appreciate the time you took to write the entire answer above. I have read it twice and I think I need more time to digest it.

 I would like to report the findings that I have obtained but by stating that they would need to be replicated. I have also calculated the effect size, which is rather small.

Rabin