  We study the classical Laplace asymptotic expansion of $\int_{\mathbb R^d}
f(x)e^{-nv(x)}dx$ in high dimensions $d$. We derive an error bound to the
expansion when truncated to arbitrary order. The error bound is fully explicit
except for absolute constants, and it depends on $d$, $n$, and operator norms
of the derivatives of $v$ and $f$ in a neighborhood of the minimizer of $v$.
