We present the discovery and extensive early-time observations of the Type Ic supernova (SN) PTF12gzk. Our light curves show a rise of 0.8 mag within 2.5 hr. Power-law fits (f(t)vprop(t - t $_0$)$^ n $) to these data constrain the explosion date to within one day. We cannot rule out a quadratic fireball model, but higher values of n are possible as well for larger areas in the fit parameter space. Our bolometric light curve and a dense spectral sequence are used to estimate the physical parameters of the exploding star and of the explosion. We show that the photometric evolution of PTF12gzk is slower than that of most SNe Ic. The high ejecta expansion velocities we measure (åisebox-0.5ex 30, 000 km s$^-1$ derived from line minima four days after explosion) are similar to the observed velocities of broad-lined SNe Ic associated with gamma- ray bursts (GRBs) rather than to normal SN Ic velocities. Yet, this SN does not show the persistent broad lines that are typical of broad-lined SNe Ic. The host-galaxy characteristics are also consistent with GRB-SN hosts, and not with normal SN Ic hosts. By comparison with the spectroscopically similar SN 2004aw, we suggest that the observed properties of PTF12gzk indicate an initial progenitor mass of 25-35 M $_☉$ and a large ((5-10) × 10$^51$ erg) kinetic energy, the later being close to the regime of GRB-SN properties.