The cooling rate of young neutron stars gives direct insight into their internal makeup. Using Chandra observations of the 330-year-old Cassiopeia A supernova remnant, we find that the temperature of the youngest-known neutron star in the Galaxy has declined by 4% over the last 10 years. The decline is explained naturally by superconductivity and superfluidity of the protons and neutrons in the stellar core. The protons became superconducting early in the life of the star and suppressed the early cooling rate; the neutron star thus remained hot before the (recent) onset of neutron superfluidity. Once the neutrons became superfluid, the Cooper pair-formation process produced a splash of neutrino emission which accelerated the cooling and resulted in the observed rapid temperature decline. This is the first time a young neutron star has been seen to cool in real time, and is the first direct evidence, from cooling observations, of superfluidity and superconductivity in the core of neutron stars.