What about exponential distribution, exp(-at) ?

If at each small time period the probability, let say, to die is constant, then this distribution will describe the durations of survival.

Please be aware of the censorship.

You may use Burr type XII distribution where its curve can cover the curve shape characteristics for different distributions including normal,exponential, Weibull,logistic, log normal and extreme value type I distribution (see Wangetal.1996).

You might want to check out the generalized gamma. The exponential, Weibull and log normal, which put certain constraints on either the relative times or relative hazards, are special cases of the GG, which is more flexible.

Cox, Christopher, et al. "Parametric survival analysis and taxonomy of hazard functions for the generalized gamma distribution." Statistics in medicine 26.23 (2007): 4352-4374.

i think exponential distribution should do the magic

You may use Weibull distribution or other Extreme Value Distributions

There is a universal model appropriate to model any data sequence. The idea is to find an optimal universal model between a random variable and the variable  you are interested in. No single distribution is needed.  but a universal use of a set of models.

Would you please tell a bit more about "the survival times follow a normal distribution"?

A priori there is no best distribution but let's say that there is a number of candidate distributions (exponential, Weibull, etc...) among which you have to select the one giving the best fit. The Normal distribution is certainly not among these. You can have a look at NICE guidelines for further explanation:

http://www.nicedsu.org.uk/Survival-analysis-TSD%282892878%29.htm