Although stars form from clouds of gas and dust, there are insignificant amounts of gas around ordinary (Sun-like) stars. This suggests that hydrogen and helium, the primary constituents of planets such as Jupiter and Saturn, are not easily retained in orbit as a star matures. The gas-giant planets in the Solar System must therefore have formed rapidly. Models of their formation generally suggest that a solid core formed in < or = 10(6) yr, followed by the accretion of the massive gaseous envelope in approximately 10(7) yr (refs 1-5). But how and when the gas of the solar nebula dissipated, and how this compares with the predicted timescale of gas-giant formation, remains unclear, in part because direct observations of circumstellar gas have been made only for stars either younger or older than the critical range of 10(6)-10(7) yr (refs 8-15). Here we report observations of the molecular gas surrounding 20 stars whose ages are likely to be in this range. The gas dissipates rapidly; after a few million years the mass remaining is typically much less than the mass of Jupiter. Thus, if gas-giant planets are common in the Galaxy, they must form even more quickly than present models suggest.