We study the following conjecture. Conjecture. Let $T$ be an $\omega$-stable theory with continuum many countable models. Then either i) $T$ has continuum many complete extensions in $L_1$, or ii) some complete extension of $T$ in $L_1$ has continuum many $L_1$-types without parameters. By Shelah's proof of Vaught's conjecture for $\omega$-stable theories, we know that there are seven types of $\omega$-stable theory with continuum many countable models. We show that the conjecture is true for all but one of these seven (...) cases. In the last case we show the existence of continuum many $L_2$-types. (shrink)