  Let $M$ be a smooth manifold, $\cal S$ the space of polynomial on fibers
functions on $T^*M$ (i.e., of symmetric contravariant tensor fields). We
compute the first cohomology space of the Lie algebra, $Vect(M)$, of vector
fields on $M$ with coefficients in the space of linear differential operators
on $\cal S$. This cohomology space is closely related to the $Vect(M)$-modules,
${\cal D}_\lambda(M)$, of linear differential operators on the space of tensor
densities on $M$ of degree $\lambda$.
