Since its discovery as a pulsar in 2000, the central compact object (CCO) 1E 1207.4-5209 in the supernova remnant PKS 1209-51/52 had been a stable 0.424 s rotator with an extremely small spin-down rate and weak (Bs ~ 9E10 G) surface dipole magnetic field. In 2016 we observed a glitch from 1E 1207.4-5209 of at least Delta f/f = (2.8+/-0.4)E-9, which is typical in size for the general pulsar population. However, glitch activity is closely correlated with spin-down rate fdot, and pulsars with fdot as small as that of 1E 1207.4-5209 are never seen to glitch. Unlike in glitches of ordinary pulsars, there may have been a large increase in fdot as well. The thermal X-ray spectrum of 1E 1207.4-5209, with its unique cyclotron absorption lines that measure the surface magnetic field strength, did not show any measurable change after the glitch, which rules out a major disruption in the dipole field as a cause or result of the glitch. A leading theory of the origin and evolution of CCOs, involving prompt burial of the magnetic field by fall-back of supernova ejecta, might hold the explanation for the glitch.