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Let $E/F$ be a quadratic extension of non-archimedean local fields ofcharacteristic 0 and let $G=U(n)$, $H=U(m)$ be unitary groups of hermitianspaces $V$ and $W$. Assume that $V$ contains $W$ and that the orthogonalcomplement of $W$ is a quasisplit hermitian space (i.e. whose unitary group isquasisplit over $F$). Let $\pi$ and $\sigma$ be smooth irreduciblerepresentations of $G(F)$ and $H(F)$ respectively. Then Gan, Gross and Prasadhave defined a multiplicity $m(\pi,\sigma)$ which for $m=n-1$ is just thedimension of $Hom_{H(F)}(\pi,\sigma)$. For $\pi$ and $\sigma$ tempered, westate and prove an integral formula for this multiplicity. As a consequence,assuming some expected properties of tempered $L$-packets, we prove a part ofthe local Gross-Prasad conjecture for tempered representations of unitarygroups. This article represents a straight continuation of recent papers ofWaldspurger dealing with special orthogonal groups.

La conjecture locale de Gross-Prasad pour les repr\'esentations temp\'er\'ees des groupes unitaires