  A detailed analysis of the most relevant sources of phase noise in an atomic
interferometer is carried out, both theoretically and experimentally. Even a
short interrogation time of 100 ms allows our cold atom gravimeter to reach an
excellent short term sensitivity to acceleration of $1.4\times 10^{-8}$g at 1s.
This result relies on the combination of a low phase noise laser system,
efficient detection scheme and good shielding from vibrations. In particular,
we describe a simple and robust technique of vibration compensation, which is
based on correcting the interferometer signal by using the AC acceleration
signal measured by a low noise seismometer.
