  CMC surfaces in spheres are investigated under the extra condition of
biharmonicity. From the work of Miyata, especially in the flat case, we give a
complete description of such immersions and show that for any $h\in (0,1)$
there exist CMC proper-biharmonic planes and cylinders in $\sn^5$ with $|H|=h$,
while a necessary and sufficient condition on $h$ is found for the existence of
CMC proper-biharmonic tori in $\sn^5$.
