We present a deep observation with the X-Ray Multimirror Mission of PSR B1823-13, a young pulsar with similar properties to the Vela pulsar. We detect two components to the X-ray emission associated with PSR B1823-13: an elongated core of extent 30'' immediately surrounding the pulsar embedded in a fainter, diffuse component of emission 5' in extent, seen only on the southern side of the pulsar. The pulsar itself is not detected, either as a point source or through its pulsations. Both components of the X-ray emission are well fitted by a power-law spectrum, with photon index Γ ≈ 1.6 and X-ray luminosity (0.5-10 keV) L_X ≈ 9 × 10³² ergs s^-1 for the core and Γ ≈ 2.3 and L_X ≈ 3 × 10³³ ergs s^-1 for the diffuse emission, for a distance of 4 kpc. We interpret both components of emission as corresponding to a pulsar wind nebula, which we designate G18.0-0.7. We argue that the core region represents the wind termination shock of this nebula, while the diffuse component indicates the shocked downstream wind. We propose that the asymmetric morphology of the diffuse emission with respect to the pulsar is the result of a reverse shock from an associated supernova remnant, which has compressed and distorted the pulsar-powered nebula. Such an interaction might be typical for pulsars at this stage in their evolution. The associated supernova remnant is not detected directly, most likely being too faint to be seen in existing X-ray and radio observations.