Using density functional theory calculations we show that the recently synthesized superhard diamondlike BC5 is superconducting with a critical temperature of the same order as that of MgB2. The average electron-phonon coupling is lambda=0.89, the phonon-frequency logarithmic average is <omega>log=67.4 meV, and the isotope coefficients are alpha(C)=0.3 and alpha(B)=0.2. In BC5, superconductivity is mostly sustained by concerted vibrations of the B atom and its C neighbors.