We present photometry of four transits of the exoplanet WASP-4b, each with a precision of approximately 500 ppm and a time sampling of 40-60 s. We have used the data to refine the estimates of the system parameters and ephemerides. During two of the transits we observed a short-lived, low-amplitude anomaly that we interpret as the occultation of a starspot by the planet. We also found evidence for a pair of similar anomalies in previously published photometry. The recurrence of these anomalies suggests that the stellar rotation axis is nearly aligned with the orbital axis, or else the starspot would not have remained on the transit chord. By analyzing the timings of the anomalies we find the sky-projected stellar obliquity to be Delta *l = --1+14 -- 12 degrees. This result is consistent with (and more constraining than) a recent observation of the Rossiter-McLaughlin effect. It suggests that the planet migration mechanism preserved the initially low obliquity, or else that tidal evolution has realigned the system. Future applications of this method using data from the CoRoT and Kepler missions will allow spin-orbit alignment to be probed for many other exoplanets.