Some planetary systems harbour debris disks containing planetesimals such as asteroids and comets. Collisions between such bodies produce small dust particles, the spectral features of which reveal their composition and, hence, that of their parent bodies. A measurement of the composition of olivine crystals (Mg2-2xFe2xSiO4) has been done for the protoplanetary disk HD 100546 (refs 3, 4) and for olivine crystals in the warm inner parts of planetary systems. The latter compares well with the iron-rich olivine in asteroids (x ~ 0.29). In the cold outskirts of the β Pictoris system, an analogue to the young Solar System, olivine crystals were detected but their composition remained undetermined, leaving unknown how the composition of the bulk of Solar System cometary olivine grains compares with that of extrasolar comets. Here we report the detection of the 69-micrometre-wavelength band of olivine crystals in the spectrum of β Pictoris. Because the disk is optically thin, we can associate the crystals with an extrasolar proto-Kuiper belt a distance of 15-45 astronomical units from the star (one astronomical unit is the Sun-Earth distance), determine their magnesium-rich composition (x = 0.01 +/- 0.001) and show that they make up 3.6 +/- 1.0 per cent of the total dust mass. These values are strikingly similar to those for the dust emitted by the most primitive comets in the Solar System, even though β Pictoris is more massive and more luminous and has a different planetary system architecture.