Aims: The association of Type Ic supernovae (SNe) with long-duration gamma-ray bursts (GRBs) is well established. We endeavor, through accurate ground-based observational campaigns, to characterize these SNe at increasingly high redshifts. Methods: We obtained a series of optical photometric and spectroscopic observations of the Type Ic SN 2012bz associated with the Swift long-duration GRB 120422A (redshift z = 0.283) using the 3.6-m TNG and the 8.2-m VLT telescopes during the time interval between 4 and 36 days after the burst. Results: The peak times of the light curves of SN 2012bz in various optical filters differ, with the B-band and i'-band light curves reaching maximum at 9 ± 4 and 23 ± 3 rest-frame days, respectively. The bolometric light curve has been derived from individual bands photometric measurements, but no correction for the unknown contribution in the near-infrared (probably around 10-15%) has been applied. Therefore, the present light curve should be considered as a lower limit to the actual UV-optical-IR bolometric light curve. This pseudo-bolometric curve reaches its maximum (Mbol = -18.56 ± 0.06) at 13 ± 1 rest-frame days; it is similar in shape and luminosity to the bolometric light curves of the SNe associated with z < 0.2 GRBs and more luminous than those of SNe associated with X-ray flashes (XRFs). A comparison with the model generated for the bolometric light curve of SN 2003dh suggests that SN 2012bz produced only about 15% less 56Ni than SN 2003dh, about 0.35 M&sun;. Similarly the VLT spectra of SN 2012bz, after correction for Galactic extinction and for the contribution of the host galaxy, suggest comparable explosion parameters with those observed in SN 2003dh (EK ~ 3.5 × 1052 erg, Mej ~ 7 M&sun;) and a similar progenitor mass (~25-40 M&sun;). GRB 120422A is consistent with the Epeak - Eiso and the EX,iso - Eγ,iso - Epeak relations. GRB 120422A / SN 2012bz shows the GRB-SN connection at the highest redshift so far accurately monitored both photometrically and spectroscopically. Based on observations made with the Italian 3.6-m Telescopio Nazionale Galileo (TNG), operated on the island of La Palma by the Fundación Galileo Galilei of the INAF (Instituto Nazionale di Astrofisica) at the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias under program A25TAC_11, and with the ESO 8.2-m Very Large Telescope (VLT) at Paranal Observatory under program 089.D-0033(A).