  We study the empirical measure $L_{A_n}$ of the eigenvalues of non-normal
square matrices of the form $A_n=U_nD_nV_n$ with $U_n,V_n$ independent Haar
distributed on the unitary group and $D_n$ real diagonal. We show that when the
empirical measure of the eigenvalues of $D_n$ converges, and $D_n$ satisfies
some technical conditions, $L_{A_n}$ converges towards a rotationally invariant
measure on the complex plan whose support is a single ring. In particular, we
provide a complete proof of Feinberg-Zee single ring theorem \cite{FZ}. We also
consider the case where $U_n,V_n$ are independent Haar distributed on the
orthogonal group.
