  Let ${S_t^H, t \geq 0} $ be a linear combination of a Brownian motion and of
an independent sub-fractional Brownian motion with Hurst index $0 < H < 1$. Its
main properties are studied and it is shown that $S^H $ can be considered as an
intermediate process between a sub-fractional Brownian motion and a mixed
fractional Brownian motion. Finally, we determine the values of $H$ for which
$S^H$ is not a semi-martingale.
