At 1.55 ms the period (P) of the newly discovered pulsar PSR 1937+21 in 4C21.53 (ref. 1) is two or three orders of magnitude smaller than most other pulsars. We have measured the rate of change of period and found it to be also remarkably small, so that the pulsar cannot be on the same evolutionary track as any other previously known. Our measured value of Ṗ is (1.7 +/- 0.7) × 10-19, which is four orders of magnitude below the previously quoted upper limit2. The formal error depends heavily on the accuracy of the assumed pulsar position: moreover, recent work suggests that there may be a systematic error which affects this position, and the correct value of Ṗ is probably as low as (0.4 +/- 0.7) × 10-19. Note therefore that even our present measurements may ultimately be providing an upper limit for Ṗ. We present here measured characteristics of this pulsar which suggests that it is nevertheless behaving like many others. The lifetime is at least 108 yr, and the magnetic dipole moment must be a factor of about 104 lower than in the Crab Pulsar. The pulsar is not a member of a binary system.