The cooling rate of young neutron stars (NSs) gives direct insight into their internal makeup. Although the temperatures of several young NSs have been measured, until now a young NS has never been observed to decrease in temperature over time. We fit nine years of archival Chandra ACIS spectra of the likely NS in the {approx}330 yr old Cassiopeia A supernova remnant with our non-magnetic carbon atmosphere model. Our fits show a relative decline in the surface temperature by 4% (5.4{sigma}, from (2.12 {+-} 0.01) x 10{sup 6} K in 2000 to (2.04 {+-} 0.01) x 10{sup 6} K in 2009) and the observed flux by 21%. Using a simple model for NS cooling, we show that this temperature decline could indicate that the NS became isothermal sometime between 1965 and 1980, and constrains some combinations of neutrino emission mechanisms and envelope compositions. However, the NS is likely to have become isothermal soon after formation, in which case the temperature history suggests episodes of additional heating or more rapid cooling. Observations over the next few years will allow us to test possible explanations for the temperature evolution.