  In this paper we produce families of Riemannian metrics with positive
constant $\sigma_k$-curvature equal to $2^{-k} {n \choose k}$ by performing the
connected sum of two given compact {\em non degenerate} $n$--dimensional
solutions $(M_1,g_1)$ and $(M_2,g_2)$ of the (positive) $\sigma_k$-Yamabe
problem, provided $2 \leq 2k < n$. The problem is equivalent to solve a second
order fully nonlinear elliptic equation.
