We study conformally flat hypersurfaces $fcolon M^{3} to Q^{4}(c)$ 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 $H eq 0$. Our main results are for the minimal case $H=0$. If $c eq 0$, we prove that if $fcolon M^{3} to Q^{4}(c)$ 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 $tilde cgeq c$ if $c>0$. For $c=0$, we show that, besides the cone over the Clifford torus in $Sf^3subset R^4$, there exists precisely a one-parameter family of (congruence classes of) minimal isometric immersions $fcolon M^3 to R^4$ with three distinct principal curvatures of simply-connected conformally flat Riemannian manifolds.
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