The young 21,000 yr old neutron star PSR B1823-13 was imaged by both the PSPC and HRI aboard ROSAT during 1993 and 1994. The data reveal that the morphology of PSR B1823-13 is similar to other neutron stars of a similar age. The pointlike neutron star component is found embedded in a compact nebula of radius similar to 20 '' and surrounded by a low surface brightness diffuse emission region of extent similar to 4'-5'. The diffuse extended component is consistent with the size expected for an associated remnant given the observed size of the compact nebula. Alternatively, the extended component may be a pulsar wind nebula and, if so, PSR B1823-13 cannot be a high-velocity neutron star unless it is embedded in an unusually rarefied region of the interstellar medium. Limits on the surface temperature of the bare neutron star are derived and compared with current models.