The discovery(1) of a planet around the solar-type star 51 Pegasi marked a watershed in the search for extrasolar planets. Since then, seven other planets have been discovered(2-6), of which several have surprisingly short orbital periods, like the planet around 51 peg. These planets were detected using the indirect technique of measuring variations in the Doppler shifts of lines in the spectra of the primary stars, But it is possible that regular oscillations of the stars themselves (or other effects) could mimic the signature of the planets, particularly the short-period planets, The apparent lack of spectral(7) and brightness(8) variations, however, led to widespread acceptance that there is a planet around 51 Peg. This conclusion was challenged by the observation(9) of systematic variations in the line shapes of 51 Peg, which suggest stellar oscillations(10). If these observations are correct, then there is no need to invoke a planet around 51 Peg to explain the data. Here we report observations of 51 Peg at a much higher spectral resolution than those in ref. 9, in which we find no evidence for systematic changes in the line shapes. The data are most consistent with a planetary companion to 51 Peg.