We present Hubble Space Telescope (HST) observations of the exceptionally bright and luminous Swift gamma-ray burst, GRB 130427A. At z=0.34 this burst affords an excellent opportunity to study the supernova and host galaxy associated with an intrinsically extremely luminous burst (E(sub iso) greater than 10(exp 54) erg): more luminous than any previous GRB with a spectroscopically associated supernova. We use the combination of the image quality, UV capability and and invariant PSF of HST to provide the best possible separation of the afterglow, host and supernova contributions to the observed light approximately 17 rest-frame days after the burst utilising a host subtraction spectrum obtained 1 year later. Advanced Camera for Surveys (ACS) grism observations show that the associated supernova, SN 2013cq, has an overall spectral shape and luminosity similar to SN 1998bw (with a photospheric velocity, vph approximately 15,000 kilometers per second). The positions of the bluer features are better matched by the higher velocity SN 2010bh (vph approximately 30,000 kilometers per second), but SN 2010bh (vph approximately 30,000 kilometers per second 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 approximately 4 kpc from the nucleus of a moderately star forming (1 Solar Mass yr(exp-1)), possibly interacting disc 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 GRB980425SN 1998bw. The similarity of supernovae and environment from both the most luminous and least luminous GRBs suggests broadly similar progenitor stars can create GRBs across six orders of magnitude in isotropic energy.