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Observations of PSR B1823-13 with the ROSAT PSPC and HRI are reported. With a period of $\sim$ 101 ms and $\dot{P}$ of $\sim$ 75$\times$10$^{-15}$ ss$^{-1}$, PSR B1823-13 has a spindown age of $\sim$ 21,000 yrs. These spin parameters yield a spindown energy loss rate, ${\dot E}$, of $\sim$ 10$^{36}$ erg s$^{-1}$. There is no known radio band supernova remnant (SNR) associated with PSR B1823-13 which calls into question either the lifetimes of SNR or the origin of this neutron star. We will address the possibility that PSR B1823-13 is imbedded in a compact nebula as observed in the Vela pulsar with the high spatial resolution HRI data. Results of a spectral analysis of the PSPC data are presented and the implications with regard to the thermal evolution of young neutron stars will be discussed.