We present Hubble Space Telescope (HST) observations of the exceptionally bright and luminous Swift gamma-ray burst (GRB), GRB 130427A. At z = 0.34, this burst affords an excellent opportunity to study the supernova (SN) and host galaxy associated with an intrinsically extremely luminous burst (E iso > 10 54 erg): more luminous than any previous GRB with a spectroscopically associated SN. We use the combination of the image quality, UV capability, and invariant point-spread function of HST to provide the best possible separation of the afterglow, host, and SN contributions to the observed light ∼17 rest-frame days after the burst, utilizing a host subtraction spectrum obtained one year later. Advanced Camera for Surveys grism observations show that the associated SN, SN 2013cq, has an overall spectral shape and luminosity similar to SN 1998bw (with a photospheric velocity, v ph ∼ 15,000 km s −1 ). The positions of the bluer features are better matched by the higher velocity SN 2010bh (v ph ∼ 30,000 km s −1 ), but this SN is significantly fainter and fails to reproduce the overall spectral shape, perhaps indicative of velocity structure in the ejecta. We find that the burst originated ∼4 kpc from the nucleus of a moderately star forming (1 M yr −1 ), possibly interacting disk galaxy. The absolute magnitude, physical size, and morphology of this galaxy, as well as the location of the GRB within it, are also strikingly similar to those of GRB 980425/SN 1998bw. The similarity of the SNe and environment from both the most luminous and least luminous GRBs suggests that broadly similar progenitor stars can create GRBs across six orders of magnitude in isotropic energy.