  We deal with minimal surfaces in the unit sphere $S^3$, which are
one-parameter families of circles. Minimal surfaces in $\R^3$ foliated by
circles were first investigated by Riemann, and a hundred years later Lawson
constructed examples of such surfaces in $S^3$. We prove that in $S^3$ there
are only two types of minimal surfaces foliated by circles, crossing the
principal lines at a constant angle. The first type surfaces are foliated by
great circles, which are bisectrices of the principal lines, and we show that
these minimal surfaces are the well-known examples of Lawson. The second type
surfaces, which are new in the literature, are families of small circles, and
the circles are principal lines. We give a constructive formula for these
surfaces. An application to the theory of minimal foliated semi-symmetric
hypersurfaces in $\R^4$ is given.
