The most important process that allows solar-wind plasma to cross the magnetopause and enter Earth's magnetosphere is the merging between solar-wind and terrestrial magnetic fields of opposite sense-magnetic reconnection. It is at present not known whether reconnection can happen in a continuous fashion or whether it is always intermittent. Solar flares and magnetospheric substorms--two phenomena believed to be initiated by reconnection--are highly burst-like occurrences, raising the possibility that the reconnection process is intrinsically intermittent, storing and releasing magnetic energy in an explosive and uncontrolled manner. Here we show that reconnection at Earth's high-latitude magnetopause is driven directly by the solar wind, and can be continuous and even quasi-steady over an extended period of time. The dayside proton auroral spot in the ionosphere--the remote signature of high-latitude magnetopause reconnection--is present continuously for many hours. We infer that reconnection is not intrinsically intermittent; its steadiness depends on the way that the process is driven.