  We investigate the question of sharp upper bounds for the Steklov eigenvalues
of a hypersurface of revolution of the Euclidean space with two boundary
components isometric to two copies of $\mathbb{S}^{n-1}$. For the case of the
first non zero Steklov eigenvalue, we give a sharp upper bound $B_n(L)$ (that
depends only on the dimension $n \ge 3$ and the meridian length $L>0$) which is
reached by a degenerated metric $g^*$, that we compute explicitly. We also give
a sharp upper bound $B_n$ which depends only on $n$. Our method also permits us
to prove some stability properties of these upper bounds.
