I'm trying to prove the title. My textbook says the proof is similar to an exercise which asks you to prove that a countable and atomic model of a complete theory is prime. I can do the exercise, but I don't see how to adapt this proof. Here is my proof of the exercise.
Suppose a model $\mathcal{A}$ of $T$ is atomic and countable and let $\mathcal{M}$ be any model of $T$. Let $a_1, a_2, \dots$ be an enumeration of $\mathcal{A}$. We want to construct a mapping $\pi \colon \mathcal{A} \to \mathcal{M}$ such that, for all formulas $\varphi(x_1, x_2, \dots, x_n)$, $\mathcal{A} \models \varphi(a_1, a_2, \dots, a_n) \iff \mathcal{M} \models \varphi(\pi(a_1), \pi(a_2), \dots, \pi(a_n))$.
Proceed by induction. For $n = 0$, this holds because $T$ is complete. Suppose the statement holds for $n$. Let $\psi$ be the principal formula of $\textrm{tp}_\mathcal{A}(a_1, a_2, \dots, a_{n + 1})$ which exists because $\mathcal{A}$ is atomic. Then $\mathcal{A} \models \psi(a_1, a_2, \dots, a_{n + 1})$, so $\mathcal{A} \models \exists x \ \psi(a_1, a_2, \dots, a_n, x)$, so $\mathcal{M} \models \exists x \ \psi(\pi(a_1), \pi(a_2), \dots, \psi(a_n), x)$ by induction hypothesis. Then there exists some $m$ such that $\mathcal{M} \models \psi(\pi(a_1), \pi(a_2), \dots, \pi(a_n), m)$. Let $\pi(a_{n + 1}) := m$. Since $\psi$ was principal, $\mathcal{M}$ models any formula $\phi$ of these variables if $\mathcal{A}$ does. Since it was the principal formula of a complete type, the converse also holds. Therefore, $\pi$ is an elementary embedding and $\mathcal{A}$ is prime.
It seems like the proof could be easily adapted if I could show that, if $\mathcal{M}$ was an $\aleph_0$-saturated and countable model, then any countable model $\mathcal{A}$ could be embedded into $\mathcal{M}$. Then I could use the fact the model is saturated to show that the embedding was elementary.
However, as it is I'm not sure how to embed the model. For instance, let $a_1, a_2, \dots$ be an enumeration of $\mathcal{A}$. What should $a_1$ map to? It's tempting to map it to some random element, but this shouldn't work intuitively, for example if the element we're mapping to witnesses some constant of the language, but $a_1$ doesn't. How do I fix this?