 Solution

An omega-stable countable theory is totally transcendental. The resulting definability and finite-base theorem for types says that every type over a parameter set $A$ is based on a finite tuple from $A$ and is determined by a countable choice of formulas over that tuple. For infinite $|A|=\kappa$, there are only $\kappa$ finite tuples from $A$ and only countably many formulas, soThus $T$ is $\kappa$-stable for every infinite cardinal $\kappa$. This is omega-stability implies stability in every infinite cardinal.