  Using density functional theory calculations we show that the
recently-synthesized, superhard diamond-like BC$_5$ is superconducting with a
critical temperature of the same order than that of MgB$_2$. 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 BC$_5$,
superconductivity is mostly sustained by vibrations of the B atom and its C
neighbors.
