2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/217509One 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.24 pages, 2 figures. To appear in J. Math. Pure ApplComplex VariablesAlgebraic Geometry32C22, 32E10, 32H05, 32M17 (Primary); 14H55 (Secondary)text