  In this paper we define an algebra structure on the vector space $L(\Sigma)$
generated by links in the manifold $\Sigma \times [0,1]$ where $\Sigma $ is an
oriented surface. This algebra has a filtration and the associated graded
algebra $L_{Gr}(\Sigma)$ is naturally a Poisson algebra. There is a Poisson
algebra homomorphism from the algebra of chord diagrams $ch(\Sigma)$ on $\Sigma
$ to $L_{Gr}(\Sigma)$.
  We show that multiplication in $L(\Sigma)$ provides a geometric way to define
a deformation quantization of the algebra of chord diagrams, provided there is
a universal Vassiliev invariant for links in $\Sigma\times [0,1]$. The
quantization descends to a quantization of the moduli space of flat connections
on $\Sigma $ and it is universal with respect to group homomorphisms. If
$\Sigma $ is compact with free fundamental group we construct a universal
Vassiliev invariant.
