  The stabilization of lasers on ultra-stable optical cavities by the
Pound-Drever-Hall (PDH) technique is a widely used method. The PDH method
relies on the phase-modulation of the laser, which is usually performed by an
electro-optic modulator (EOM). When approaching the $10^{-16}$ level, this
technology requires an active control of the residual amplitude modulation
(RAM) generated by the EOM in order to bring the frequency stability of the
laser down to the thermal noise limit of the ultra-stable cavity. In this
article, we report on the development of an active system of RAM reduction
based on a free space EOM, which is used to perform PDH-stabilization of a
laser on a cryogenic silicon cavity. A RAM stability of $1.4 \times 10^{-7}$ is
obtained by employing a digital servo that stabilizes the EOM DC electric
field, the crystal temperature and the laser power. Considering an ultra-stable
cavity with a finesse of $2.5\times 10^5$, this RAM level would contribute to
the fractional frequency instability at the level of about $5\times 10^{-19}$,
well below the state of the art thermal noise limit of a few $ 10^{-17}$.
