We report on the discovery and characterization of the transiting planet K2-39b (EPIC 206247743b). With an orbital period of 4.6 days, it is the shortest-period planet orbiting a subgiant star known to date. Such planets are rare, with only a handful of known cases. The reason for this is poorly understood but may reflect differences in planet occurrence around the relatively high-mass stars that have been surveyed, or may be the result of tidal destruction of such planets. K2-39 (EPIC 206247743) is an evolved star with a spectroscopically derived stellar radius and mass of {3.88}-0.42+0.48 {R}ȯ and {1.53}-0.12+0.13 {M}ȯ , respectively, and a very close-in transiting planet, with a/{R}\star =3.4. Radial velocity (RV) follow-up using the HARPS, FIES, and PFS instruments leads to a planetary mass of {50.3}-9.4+9.7 {M}\oplus . In combination with a radius measurement of 8.3+/- 1.1 {R}\oplus , this results in a mean planetary density of {0.50}-0.17+0.29 g cm‑3. We furthermore discover a long-term RV trend, which may be caused by a long-period planet or stellar companion. Because K2-39b has a short orbital period, its existence makes it seem unlikely that tidal destruction is wholly responsible for the differences in planet populations around subgiant and main-sequence stars. Future monitoring of the transits of this system may enable the detection of period decay and constrain the tidal dissipation rates of subgiant stars. Based on observations made with the NOT telescope under program ID. 50-022/51-503, 50-213(CAT), 52-201 (CAT), 52-108 (OPTICON), 51-211 (CAT), and ESOs 3.6 m telescope at the La Silla Paranal Observatory under program ID 095.C-0718(A).