We present the analysis of a peculiar W Virginis (pWVir) type II Cepheid, OGLE-LMC-T2CEP-211 (P-puls = 9.393 days), in a double-lined binary system (P-orb = 242 days), which shed light on virtually unknown evolutionary status and structure of pWVir stars. The dynamical mass of the Cepheid (first ever for a type II Cepheid) is 0.64 +/- 0.02M(circle dot), and the radius R = 25.1 +/- 0.3 R circle dot. The companion is a massive (5.67M(circle dot)) main-sequence star obscured by a disk. Such a configuration suggests a mass transfer in the system history. We found that originally the system (P-orb(init) = 12 days) was composed of 3.5 and 2.8M(circle dot) stars, with the current Cepheid being more massive. The system age is now similar to 200 Myr, and the Cepheid is almost completely stripped of hydrogen, with helium mass of similar to 92% of the total mass. It finished transferring the mass 2.5 Myr ago and is evolving toward lower temperatures passing through the instability strip. Comparison with observations indicates a reasonable 2.7 x 10(-8)M(circle dot) yr(-1) mass loss from the Cepheid. The companion is most probably a Be main-sequence star with T = 22,000 K and R = 2.5 R-circle dot. Our results yield a good agreement with a pulsation theory model for a hydrogen-deficient pulsator, confirming the described evolutionary scenario. We detected a two-ring disk (R-disk similar to 116 R-circle dot) and a shell (R-shell similar to 9 R-circle dot) around the companion, which is probably a combination of the matter from the past mass transfer, the mass being lost by the Cepheid owing to wind and pulsations, and a decretion disk around a rapidly rotating secondary. Our study, together with observational properties of pWVir stars, suggests that the majority of them are products of a similar binary evolution interaction.