If Q is a real, symmetric and positive definite nxn matrix, and
B a real nxn matrix whose eigenvalues have negative real parts,
we consider the Ornstein--Uhlenbeck semigroup on R^n with covariance Q
and drift matrix B. Our main result is that the associated maximal operator
is of weak type (1,1) with respect to the invariant measure.
The proof has a geometric gist and hinges on the
``forbidden zones method'' previously introduced by the third author.
For large values of the time parameter,
we also prove a refinement of this result, in the spirit of a conjecture due to Talagrand
