This article is a continuation of previous works by Bovier–Eckhoff–Gayrard–Klein, Bovier–Gayrard–Klein and Helffer–Klein–Nier. The main object is the analysis of the small eigenvalues (as $h\ar 0$) of the Laplacian attached to the quadratic form$$ C_0^\infty (\Omega )\ni v \mapsto h^2 \,\int _{\Omega }\left |\nabla v (x)\right |^2\;e^{-2f(x)/h}~dx\;, $$ where $\Omega $ is a bounded connected open set with $C^\infty $-boundary and $f$ is a Morse function on $M=\overline {\Omega }$. The previous works were devoted to the case of a manifold $M$ which is compact but without boundary or $\mathbb{R}^n$. Our aim is here to analyze the case with boundary. After the introduction of a Witten cohomology complex adapted to the case with boundary, we give a very accurate asymptotics for the exponentially small eigenvalues. In particular, we analyze the effect of the boundary in the asymptotics.