We compare the chemical abundances at the sites of 12 nearby (z <0.14) Type Ic supernovae (SN Ic) that showed broad lines, but had no observed Gamma-Ray Burst (GRB), with the chemical abundances in 5 nearby (z < 0.25) galaxies at the sites of GRB where broad-lined SN Ic were seen after the fireball had faded. It has previously been noted that GRB hosts are low in luminosity and low in their metal abundances. If low metallicity is sufficient to force the evolution of massive stars to end their lives as GRB with an accompanying broad-lined SN Ic, then we would expect higher metal abundances for the broad-lined SN Ic that have no detected GRB. This is what we observe, and this trend is independent of the choice of metallicity calibration we adopt, and the mode of SN survey that found the broad-lined SN. A unique feature of this analysis is that we present new spectra of the host galaxies and analyze all the measurements of both samples in the same set of ways, using the galaxy emission-line measurements corrected for extinction and stellar absorption, via independent metallicity diagnostics of Kewley & Dopita (2002), of McGaugh (1991) and of Pettini & Pagel (2004). In our small sample, the boundary between galaxies that have GRB accompanying their broad-lined SN Ic and those that have broad-lined SN Ic without a GRB lies at an oxygen abundance of 12+log(O/H)_KD02 ~ 8.5, which corresponds to 0.2-0.6 Z_sun depending on the adopted metallicity scale and solar abundance value. Even when we limit the comparison to SN Ic that were found in untargeted supernova surveys, the environment of every broad-lined SN Ic that had no GRB is more metal rich than the site of any broad-lined SN Ic where a GRB was detected.