The duration of a gamma-ray burst (GRB) is a key indicator of its physical origin, with long bursts perhaps associated with the collapse of massive stars and short bursts with mergers of neutron stars. However, there is substantial overlap in the properties of both short and long GRBs and neither duration nor any other parameter so far considered completely separates the two groups. Here we unambiguously classify every GRB using a machine-learning dimensionality reduction algorithm, t-distributed stochastic neighborhood embedding, providing a catalog separating all Swift GRBs into two groups. Although the classification takes place only using prompt emission light curves, every burst with an associated supernova is found in the longer group and bursts with kilonovae in the short, suggesting along with the duration distributions that these two groups are truly long and short GRBs. Two bursts with a clear absence of a supernova belong to the longer class, indicating that these might have been direct-collapse black holes, a proposed phenomenon that may occur in the deaths of more massive stars.