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