Given a plane graph G (i.e., a planar graph with a fixed planar embedding)
and a simple cycle C in G whose vertices are mapped to a convex polygon, we
consider the question whether this drawing can be extended to a planar
straight-line drawing of G. We characterize when this is possible in terms of
simple necessary conditions, which we prove to be sufficient. This also leads
to a linear-time testing algorithm. If a drawing extension exists, it can be
computed in the same running time