The main result is Vaught's conjecture for weakly minimal, locally modular and non-$\omega$-stable theories. The more general results yielding this are the following. THEOREM A. Suppose that $T$ is a small unidimensional theory and $D$ is a weakly minimal set, definable over the finite set $B$. Then for all finite $A \subset D$ there are only finitely many nonalgebraic strong types over $B$ realized in $\operatorname{acl}(A) \cap D$. THEOREM B. Suppose that $T$ is a small, unidimensional, non-$\omega$-stable theory such that the universe is weakly minimal and locally modular. Then for all finite $A$ there is a finite $B \subset \mathrm{cl}(A)$ such that $a \in \mathrm{cl}(A)$ iff $a \in \mathrm{cl}(b)$ for some $b \in B$. Recall the property (S) defined in the abstract of [B1]. THEOREM C. Let $T$ be as in Theorem B. Then, if $T$ does not satisfy (S), $T$ has $2^{\aleph_0}$ many countable models. Combining Theorem C and the results in [B1] we obtain Vaught's conjecture for such theories.