  We prove a new weak mean ergodic theorem (Theorem A) for 1-cocycles
associated to weakly mixing representations of amenable groups. Let $G$ be a
finitely generated, discrete, amenable group $G$ which admits a controlled
Folner sequence. We use Theorem A to deduce that any affine action $G\ca^T \Cal
H$ on Hilbert space with weakly mixing linear part admits a sequence of almost
fixed points (Theorem B). Specializing to the case that $G$ is a finitely
generated group of polynomial growth, we show that convex combinations of
averages of the associated 1-cocycle over $n$-balls provide a sequence of
almost fixed points for the action $G\ca^T \Cal H$ (Corollary C). This affirms
a weak form of a conjecture of Shalom independently of Gromov's theorem on the
virtual nilpotency of groups of polynomial growth. As a consequence, we are
able to give a new, elementary, ergodic-theoretical proof of Gromov's theorem.
