First of all, if you have a whole number of periods, the FFT of your sine wave will have no spectral leakage and no DC bias caused by the spectral leakage. That being said, if your sine wave has been acquired via an analog chain and an ADC, the DC bias could be caused by the analog input chain and the ADC.
Second, the presence of a DC bias will no affect the amplitude of the other FFT bins, therefore removing the mean will not yield a better amplitude estimate.
Third, if you don't have a whole number of periods, the FFT is probably not the right tool to estimate the amplitude of a sine wave.
Finally, I don't know how your gaussian noise was generated. It should be zero-mean. That being said, it's possible that there could be some DC bias especially if your noise sequence is short.