We present near and mid-infrared observations of the pulsar-wind nebula (PWN) B0540-69.3 and its associated supernova remnant made with the {it Spitzer Space Telescope}. We report detections of the PWN with all four IRAC bands, the 24 $mu$m band of MIPS, and the Infrared Spectrograph (IRS). We find no evidence of IR emission from the X-ray/radio shell surrounding the PWN resulting from the forward shock of the supernova blast wave. The flux of the PWN itself is dominated by synchrotron emission at shorter (IRAC) wavelengths, with a warm dust component longward of 20 $mu$m. We show that this dust continuum can be explained by a small amount ($sim 1-3 times 10^{-3} msun$) of dust at a temperature of $sim 50-65$ K, heated by the shock wave generated by the PWN being driven into the inner edge of the ejecta. This is evidently dust synthesized in the supernova. We also report the detection of several lines in the spectrum of the PWN, and present kinematic information about the PWN as determined from these lines. Kinematics are consistent with previous optical studies of this object. Line strengths are also broadly consistent with what one expects from optical line strengths. We find that lines arise from slow ($sim 20$ km s$^{-1}$) shocks driven into oxygen-rich clumps in the shell swept-up by an iron-nickel bubble, which have a density contrast of $sim 100-200$ relative to the bulk of the ejecta, and that faster shocks ($sim 250$ km s$^{-1}$) in the hydrogen envelope are required to heat dust grains to observed temperatures. We infer from estimates of heavy-element ejecta abundances that the progenitor star was likely in the range of 20-25 $M_odot$.
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