Recently a Type Ic supernova, SN 1998bw, was discovered coincident with a gamma-ray burst, GRB 980425. The supernova had unusual radio, optical, and spectroscopic properties. We explore here models based upon helium stars in the range 9 - 14 Msun and carbon-oxygen stars 6 - 11 Msun which experience unusually energetic explosions (kinetic energy 0.5 - 2.8 x 10**52 erg). Bolometric light curves and multi-band photometry are calculated and compared favorably with observations. Both LTE and non-LTE spectra are also calculated for the model that agrees best with the light curve, a carbon-oxygen core of 6 Msun exploded with a kinetic energy of 2.2 x 10**52 erg. We also examine potential mechanisms for producing the observed gamma-ray burst (GRB) - shock break-out and relativistic shock deceleration in circumstellar material. For spherically symmetric models, both fail to produce a GRB of even the low luminosity inferred for GRB 980425. The high explosion energies required to understand the supernova are in contrast to what is expected for such massive stars and may indicate that a new sort of explosion has been identified, possibly the consequence of a collapsar (Woosley 1993, 1996). Indeed a more likely explanation for what was seen is a highly asymmetric explosion in which the GRB was produced by a relativistic jet, perhaps viewed obliquely, and only a fraction of the total stellar mass was ejected, the remainder accreting into a black hole. The ejected mass (but not the 56Ni mass), explosion energy, and velocities may then be smaller. Other Type Ic supernova - GRB coincidences are suggested.

24 pages including 6 figures, submitted to The Astrophysical Journal. latex uses aaspp4.sty