  Let $X$ be a smooth projective variety over a field of characteristic $0$ and
$L$ an ample divisor. In this paper we study the Weak Lefschetz conjecture for
Chow groups using the technique employed by Grothendieck in his study of the
problem for Picard groups, and using Bloch's formula to interpret Chow groups
in terms of Milnor $K$- theory we prove the infinitesimal part of the
conjecture.
