We present the analysis of a peculiar W Virginis (pWVir) type II Cepheid, OGLE-LMC-T2CEP-211 (P_puls_=9.393d), in a double-lined binary system (P_orb_=242d), 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_{sun}_, and the radius R=25.1+/-0.3R_{sun}_. The companion is a massive (5.67M_{sun}_) 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^=12d) was composed of 3.5 and 2.8M_{sun}_ stars, with the current Cepheid being more massive. The system age is now ~200Myr, and the Cepheid is almost completely stripped of hydrogen, with helium mass of ~92% of the total mass. It finished transferring the mass 2.5Myr ago and is evolving toward lower temperatures passing through the instability strip. Comparison with observations indicates a reasonable 2.7x10^-8^M_{sun}_/yr mass loss from the Cepheid. The companion is most probably a Be main-sequence star with T=22000K and R=2.5R_{sun}_. 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_~116R_{sun}_) and a shell (R_shell_~9R_{sun}_) 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.