  We study conformally flat hypersurfaces $f\colon M^{3} \to \Q^{4}(c)$ with
three distinct principal curvatures and constant mean curvature $H$ in a space
form with constant sectional curvature $c$. First we extend a theorem due to
Defever when $c=0$ and show that there is no such hypersurface if $H\neq 0$.
Our main results are for the minimal case $H=0$. If $c\neq 0$, we prove that if
$f\colon M^{3} \to \Q^{4}(c)$ is a minimal conformally flat hypersurface with
three distinct principal curvatures then $f(M^3)$ is an open subset of a
generalized cone over a Clifford torus in an umbilical hypersurface
$\Q^{3}(\tilde c)\subset \Q^4(c)$, $\tilde c>0$, with $\tilde c\geq c$ if
$c>0$. For $c=0$, we show that, besides the cone over the Clifford torus in
$\Sf^3\subset \R^4$, there exists precisely a one-parameter family of
(congruence classes of) minimal isometric immersions $f\colon M^3 \to \R^4$
with three distinct principal curvatures of simply-connected conformally flat
Riemannian manifolds.
