Context. The evolution of planetary systems is intimately linked to the evolution of their host stars. Our understanding of the whole planetary evolution process is based on the wide planet diversity observed so far. Only a few tens of planets have been discovered orbiting stars ascending the red giant branch. Although several theories have been proposed, the question of how planets die remains open owing to the small number statistics, making it clear that the sample of planets around post-main sequence stars needs to be enlarged. Aims: In this work we study the giant star Kepler-91 (KOI-2133) in order to determine the nature of a transiting companion. This system was detected by the Kepler Space Telescope, which identified small dims in its light curve with a period of 6.246580 ± 0.000082 days. However, its planetary confirmation is needed due to the large pixel size of the Kepler camera, which can hide other stellar configurations able to mimic planet-like transit events. Methods: We analysed Kepler photometry to 1) re-calculate transit parameters; 2) study the light-curve modulations; and 3) to perform an asteroseismic analysis (accurate stellar parameter determination) by identifying solar-like oscillations on the periodogram. We also used a high-resolution and high signal-to-noise ratio spectrum obtained with the Calar Alto Fiber-fed Échelle spectrograph (CAFE) to measure stellar properties. Additionally, false-positive scenarios were rejected by obtaining high-resolution images with the AstraLux lucky imaging camera on the 2.2 m telescope at the Calar Alto Observatory. Results: We confirm the planetary nature of the object transiting the star Kepler-91 by deriving a mass of Mp=0.88+0.17-0.33 MJup and a planetary radius of Rp=1.384+0.011-0.054 RJup. Asteroseismic analysis produces a stellar radius of R⋆ = 6.30 ± 0.16 R⊙ and a mass of M⋆ = 1.31 ± 0.10 M⊙. We find that its eccentric orbit (e=0.066+0.013-0.017) is just 1.32+0.07-0.22 R⋆ away from the stellar atmosphere at the pericentre. We also detected three small dims in the phase-folded light curve. The combination of two of them agrees with the theoretical characteristics expected for secondary eclipse. Conclusions: Kepler-91b could be the previous stage of the planet engulfment, which has recently been detected for BD+48 740. Our estimations show that Kepler-91b will be swallowed by its host star in less than 55 Myr. Among the confirmed planets around giant stars, this is the closest to its host star. At pericentre, the star subtends an angle of 48°, covering around 10% of the sky as seen from the planet. The planetary atmosphere seems to be inflated probably due to the high stellar irradiation. Based on observations collected at the German-Spanish Astronomical Center, Calar Alto, jointly operated by the Max-Planck-Institut fur Astronomie (Heidelberg) and the Instituto de Astrofísica de Andalucía (IAA-CSIC, Granada).Appendix A is available in electronic form at http://www.aanda.org