  Gromov, in his seminal 1989 paper on the Oka principle, introduced the notion
of an elliptic manifold and proved that every continuous map from a Stein
manifold to an elliptic manifold is homotopic to a holomorphic map. We show
that a much stronger Oka principle holds in the special case of maps from
certain open Riemann surfaces called circular domains into CxC*, namely that
every continuous map is homotopic to a proper holomorphic embedding. An
important ingredient is a generalisation to CxC* of recent results of Wold and
Forstneric on the long-standing problem of properly embedding open Riemann
surfaces into C^2, with an additional result on the homotopy class of the
embeddings. We also give a complete solution to a question that arises
naturally in Larusson's holomorphic homotopy theory, of the existence of
acyclic embeddings of Riemann surfaces with abelian fundamental group into
2-dimensional elliptic Stein manifolds.
