β Pictoris (β Pic) is a main-sequence star with an edge-on dust disk that might represent a state of the early Solar System. The dust does not seem to be a remnant from the original protoplanetary disk, but rather is thought to have been generated from large bodies like planetesimals and/or comets. The history and composition of the parent bodies can therefore be revealed by determining the spatial distribution, grain size, composition and crystallinity of the dust through high-resolution mid-infrared observations. Here we report that the sub-micrometre amorphous silicate grains around β Pic have peaks in their distribution around 6, 16 and 30AU (1AU is the Sun-Earth distance), whereas the crystalline and micrometre-sized amorphous silicate grains are concentrated in the disk centre. As sub-micrometre grains are blown quickly out from the system by radiation pressure from the central star, the peaks indicate the locations of ongoing dust replenishment, which originates from ring-like distributions of planetesimals or `planetesimal belts'.