We present the analysis of a peculiar W Virginis (pWVir) type-II Cepheid, OGLE-LMC-T2CEP-211 (P puls = 9.393 d), in a double-lined binary system (P orb = 242 d), 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 ± 0.02 M ⊙ and the radius R = 25.1 ± 0.3 R ⊙ . The companion is a massive (5.67 M ⊙ ) main-sequence star obscured by a disk. Such configuration suggests a mass transfer in the system history. We found that originally the system (P init orb = 12 d) was composed of 3.5 and 2.8 M ⊙ stars, with the current Cepheid being more massive. The system age is now ∼ 200 My, and the Cepheid is almost completely stripped of hydrogen, with helium mass of ∼ 92% of the total mass. It finished transferring the mass 2.5 My ago and is evolving towards lower temperatures passing through the instability strip. Comparison with observations indicate a reasonable 2.7·10 −8 M ⊙ /y mass loss from the Cepheid. The companion is most probably a Be main-sequence star with T = 22000 K and R = 2.5 R ⊙ . 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 ∼ 116 R ⊙ ) and a shell (R shell ∼ 9 R ⊙ ) around the companion, that is probably a combination of the matter from the past mass transfer, the mass being lost by the Cepheid due to wind and pulsations, and a decretion disk around a rapidly rotating secondary. Our study together with observational properties of pWVir stars suggests that their majority are products of a similar binary evolution interaction.