True rms detector provides True rms power equivalent true rms voltage. keeping load/termination constant, take measurement by switching off signal (power in absence of signal) and take again reading in presence of signal. difference is the required power. For better accuracy alternately, use sharp band narrow band filter and repeat the above process.

In case band reject filter is used, the difference is from second measurement to first measurement.  

You can only get an estimate of the true power, because, as you note, you are seeing the power of the noise and the power of the signal combined, and the noise power is a random variable.    The variance in the measurement is the variance of the noise power (which itself, is the variance of the noise voltage, so what you're really looking at is the variance of the variance).

A typical expression is that the  fractional 1 sigma uncertainty (linear scale) is 1+/- (1/SNR). At 3 dB SNR (SNR=2) the the 1 sigma range around a signal of power 1 watt is 0.5 to 1.5 Watt.

One can, of course, integrate over a longer time to reduce the contribution of the noise (the usual sqrt(N) reduction applies,as with summing any random variate).  

A review of the HP/Agilent/Keysight Ap note 64-1 might help. There's a discussion of microwave power measurement effects.  Look around to find the newest version.  64-1 doesn't really get into low SNR measurements, though.

http://rftest.blogs.keysight.com/2013/09/low-snr-measurements-how-close-to-noise-is-too-close/ has a table, but no analytical rationale.

suggest the use of a selective filter toned to the desired signal; although it is difficult to achieve quadrature band limits. through cascade arrangement, as the Chebyshev filter.

I missed that you were measuring noise power with a noisy receiver. This is straightforward radiometry. Use the Y factor method.  Measure the output of the receiver with the input connected to a load at a known temperature. Measure the output of the receiver with the input connected to the load at a different known temperature (or to a calibrated noise source).  You can calculate the gain and noise temperature of the receiver.  (you may already have done this...)  

Now measure the output of the receiver with the unknown signal connected (whether it's noise or a sine wave or any waveform).  The added power is your unknown, scaled by the gain of the receiver to refer it to the input.  My previous answer tells you how to assess the uncertainty of the measurement.

There is a good description of this process in Kraus's book on Antennas, and in his book on radio astronomy.  Measuring the noise power of things beyond your antenna is what radio astronomy is all about.  

You could also look at the Deep Space Network literature, where they take truly obsessive care with measuring the system noise temperature.  http://descanso.jpl.nasa.gov would get you started, but that's probably overkill.

Excellent, James. Thanks a lot for your explanations. This will help me a lot. Will look up the paragraphs in both books of Kraus. I have Antennas here, but maybe I should also obtain a copy of Radio Astronomy, wanted to have that on the shelve a long time.

Kindest regards!

Ben