  One of the oldest open problems in the classical function theory is whether
every open Riemann surface admits a proper holomorphic embedding into C^2. In
this paper we prove the following Theorem: If D is a bordered Riemann surface
whose closure admits an injective immersion in C^2 that is holomorphic in D,
then D admits a proper holomorphic embedding in C^2. The most general earlier
results are due to J. Globevnik and B. Stensones (Math. Ann. 303 (1995),
579-597) and E. F. Wold (Internat. J. Math. 17 (2006), 963-974). We give an
explicit and elementary construction that does not require the Teichmuller
space theory, and we also indicate another possible proof using the latter
theory.
