= Solution
Let $\lambda=|T|$ and choose an infinite cardinal $\kappa$ satisfying $\kappa^\lambda=\kappa$, for example $\kappa=2^\lambda$. Take $M\models T$ of cardinality $\kappa$. By assumption, every $p\in S_1(M)$ is definable.
For each parameter-free formula scheme $\varphi(x;y)$, the definition $d_p\varphi(y)$ is a formula with finitely many parameters from $M$. There are at most $\kappa$ such formulas. A definable type is completely determined by choosing one definition for each of the at most $\lambda$ formula schemes, so
$$
|S_1(M)|\leq\kappa^\lambda=\kappa.
$$
The same count applies to every model of cardinality $\kappa$, and therefore $T$ is $\kappa$-stable. Hence $T$ is a <stable theory>.
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