Gamma-ray bursts (GRBs) are known to come in two duration classes, separated at approximately 2 s. Long-duration bursts originate from star-forming regions in galaxies, have accompanying supernovae when these are near enough to observe and are probably caused by massive-star collapsars. Recent observations show that short-duration bursts originate in regions within their host galaxies that have lower star-formation rates, consistent with binary neutron star or neutron star-black hole mergers. Moreover, although their hosts are predominantly nearby galaxies, no supernovae have been so far associated with short-duration GRBs. Here we report that the bright, nearby GRB 060614 does not fit into either class. Its approximately 102-s duration groups it with long-duration GRBs, while its temporal lag and peak luminosity fall entirely within the short-duration GRB subclass. Moreover, very deep optical observations exclude an accompanying supernova, similar to short-duration GRBs. This combination of a long-duration event without an accompanying supernova poses a challenge to both the collapsar and the merging-neutron-star interpretations and opens the door to a new GRB classification scheme that straddles both long- and short-duration bursts.