Was GRB~980329 at z ~ 5?

The optical transient (OT) associated with GRB~980329 was remarkably red. It has previously been concluded that this was the result of dust extinction in the host galaxy. However, an extinction model can only agree with the data if the $I$ band observations taken about 0.8 days after outburst are discounted; the flux density ratio between the $I$ and the $R$ of a factor $\sim 7$ is too great to be explained by extinction, given the relatively blue $K-I$ color. Here it is shown that the {\it entire} observed optical/infrared spectrum is consistent with that which is expected from an unextincted OT at $z \sim 5$. At this redshift, the light in the observer's $R$ band is strongly suppressed by absorption in the Ly$\alpha$ forest -- an effect which has been clearly seen in galaxies in the Hubble Deep Field. In spite of its potentially high redshift, GRB~980329 was an unusually bright burst. If GRB 980329 was indeed at $z \sim 5$, and its gamma-rays were radiated isotropically, the implied energy of the burst would be $ 5 \times 10^{54}$~ergs. Should GRB~980329 have a host galaxy, deep imaging could confirm or reject the conclusion that this burst was at $z \sim 5$.