Let $X$ be an orbifold that is a global quotient of a manifold $Y$
by a finite group $G$. We construct a noncommutative ring $H\sp
\ast(Y, G)$ with a $G$-action such that $H\sp*(Y, G)\sp G$ is the
orbifold cohomology ring of $X$ defined by W. Chen and Y. Ruan
[CR]. When $Y=S\sp n$, with $S$ a surface with trivial canonical class
and $G = \mathfrak {S}\sb n$, we prove that (a small modification of)
the orbifold cohomology of $X$ is naturally isomorphic to the
cohomology ring of the Hilbert scheme $S\sp {[n]}$, computed by
M. Lehn and C. Sorger [LS2].