  We study the effect that the injection of a common source of noise has on the
trajectories of chaotic systems, addressing some contradictory results present
in the literature. We present particular examples of 1-d maps and the Lorenz
system, both in the chaotic region, and give numerical evidence showing that
the addition of a common noise to different trajectories, which start from
different initial conditions, leads eventually to their perfect
synchronization. When synchronization occurs, the largest Lyapunov exponent
becomes negative. For a simple map we are able to show this phenomenon
analytically. Finally, we analyze the structural stability of the phenomenon.
