Long-duration gamma-ray bursts (GRBs) at z < 1 are found in most cases to be accompanied by bright, broad-lined Type Ic supernovae (SNe Ic-BL). The highest-energy GRBs are mostly located at higher redshifts, where the associated SNe are hard to detect observationally. Here, we present early and late observations of the optical counterpart of the very energetic GRB 130427A. Despite its moderate redshift, z = 0.3399 ± 0.0002, GRB 130427A is at the high end of the GRB energy distribution, with an isotropic-equivalent energy release of E iso ~ 9.6 × 1053 erg, more than an order of magnitude more energetic than other GRBs with spectroscopically confirmed SNe. In our dense photometric monitoring, we detect excess flux in the host-subtracted r-band light curve, consistent with that expected from an emerging SN, ~0.2 mag fainter than the prototypical SN 1998bw. A spectrum obtained around the time of the SN peak (16.7 days after the GRB) reveals broad undulations typical of SNe Ic-BL, confirming the presence of an SN, designated SN 2013cq. The spectral shape and early peak time are similar to those of the high expansion velocity SN 2010bh associated with GRB 100316D. Our findings demonstrate that high-energy, long-duration GRBs, commonly detected at high redshift, can also be associated with SNe Ic-BL, pointing to a common progenitor mechanism.