  In this paper we study surfaces in Euclidean 3-space foliated by pieces of
circles and that satisfy a Weingarten condition of type $a H+b K=c$, where
$a,b$ and $c$ are constant and $H$ and $K$ denote the mean curvature and the
Gauss curvature respectively. We prove that a such surface must be a surface of
revolution, a Riemann minimal surface or a generalized cone.
