= Solution
Assume $M$ is a <prime model>. By the downward Lowenheim-Skolem theorem, $T$ has a countable model, and the elementary embedding of $M$ into it makes $M$ countable. If a tuple $\bar a\in M$ had a nonisolated type, the omitting types theorem would give a countable model of $T$ omitting that type. An elementary embedding of $M$ into this model would realize it, a contradiction. Thus $M$ is <atomic model>[atomic].
Conversely, let $M$ be countable and atomic, enumerate it as $(a_i)_{i<\omega}$, and let $N\models T$. Construct an elementary embedding recursively. Suppose $a_0,\ldots,a_{n-1}$ have been mapped to $\bar b$. Let $\psi(\bar x)$ isolate the type of $(a_0,\ldots,a_{n-1})$ and let $\theta(\bar x,y)$ isolate the type of $(a_0,\ldots,a_n)$. Since the latter extends the former and is realized in $M$, completeness gives
$$
T\models\forall\bar x\,
\bigl(\psi(\bar x)\to\exists y\,\theta(\bar x,y)\bigr).
$$
The tuple $\bar b$ realizes $\psi$, so a suitable image of $a_n$ exists in $N$. The union of the finite partial elementary maps is an elementary embedding $M\to N$. Hence $M$ is prime.
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