  Let $\mathcal{K}$ be the space of properly embedded minimal tori in quotients
of $\R^3$ by two independent translations, with any fixed (even) number of
parallel ends. After an appropriate normalization, we prove that $\mathcal{K}$
is a 3-dimensional real analytic manifold that reduces to the finite coverings
of the examples defined by Karcher, Meeks and Rosenberg in \cite{ka4,ka6,mr3}.
The degenerate limits of surfaces in $\mathcal{K}$ are the catenoid, the
helicoid and three 1-parameter families of surfaces: the simply and doubly
periodic Scherk minimal surfaces and the Riemann minimal examples.
