  The purpose of this paper is the numerical analysis of a first order
fractional-step time-scheme, using decomposition of theviscosity, and "inf-sup"
stable finite element space-approximations for the Primitive Equations of the
Ocean. The aim of the paper is twofold. Firstly, we prove that the scheme is
unconditionally stable and convergent towards weak solutions of the Primitive
Equations. Secondly, optimal error estimates for velocity and pressure are
provided of order $O(k+h^l)$ for $l=1$ or $l=2$ when either first or second
order finite-element approximations are considered ($k$ and $h$ being the time
step and the mesh size, respectively). In both cases, these error estimates are
obtained under the same constraint $k\le h^2$.
