I believe that the answer is yes (provided there are no further limit cycles within the first one).
Let there be a stable limit cycle with no other limit cycles within it. By reversing time, $t \rightarrow -t$, we get a new system with trajectories bounded to the interior of the limit cycle. Using, say, Brouwer’s fixed point theorem, one can show that there must be a fixed point within the region.
However, when reading different formulations of the Poincaré–Bendixson theorem, I often encountered the statements like this:
If the trajectories are bounded and there are no fixed points, then the trajectories must converge to a limit cycle.
Are these are just unlucky/incorrect formulations or there is something in it?