We present the discovery and follow-up observations of a broad-line Type Ic supernova (SN), PTF 10bzf (SN 2010ah), detected by the Palomar Transient Factory (PTF) on 2010 February 23. The SN distance is ≅218 Mpc, greater than GRB 980425/SN 1998bw and GRB 060218/SN 2006aj, but smaller than the other SNe firmly associated with gamma-ray bursts (GRBs). We conducted a multi-wavelength follow-up campaign with Palomar 48 inch, Palomar 60 inch, Gemini-N, Keck, Wise, Swift, the Allen Telescope Array, Combined Array for Research in Millimeter-wave Astronomy, Westerbork Synthesis Radio Telescope, and Expanded Very Large Array. Here we compare the properties of PTF 10bzf with those of SN 1998bw and other broad-line SNe. The optical luminosity and spectral properties of PTF 10bzf suggest that this SN is intermediate, in kinetic energy and amount of ^(56)Ni, between non-GRB-associated SNe like 2002ap or 1997ef, and GRB-associated SNe like 1998bw. No X-ray or radio counterpart to PTF 10bzf was detected. X-ray upper limits allow us to exclude the presence of an underlying X-ray afterglow as luminous as that of other SN-associated GRBs such as GRB 030329 or GRB 031203. Early-time radio upper limits do not show evidence for mildly relativistic ejecta. Late-time radio upper limits rule out the presence of an underlying off-axis GRB, with energy and wind density similar to the SN-associated GRB 030329 and GRB 031203. Finally, by performing a search for a GRB in the time window and at the position of PTF 10bzf, we find that no GRB in the interplanetary network catalog could be associated with this SN
