  Let $r \geq 2$ be a fixed integer. For infinitely many $n$, let
$\boldsymbol{k} = (k_1,..., k_n)$ be a vector of nonnegative integers such that
their sum $M$ is divisible by $r$. We present an asymptotic enumeration formula
for simple $r$-uniform hypergraphs with degree sequence $k$. (Here "simple"
means that all edges are distinct and no edge contains a repeated vertex.) Our
formula holds whenever the maximum degree $k_{\mathrm{max}}$ satisfies
$k_{\mathrm{max}}^3 = o(M)$.
