  We introduce a fast implementation of the pivot algorithm for self-avoiding
walks, which we use to obtain large samples of walks on the cubic lattice of up
to $33 \times 10^6$ steps. Consequently the critical exponent $\nu$ for
three-dimensional self-avoiding walks is determined to great accuracy; the
final estimate is $\nu=0.587597(7)$. The method can be adapted to other models
of polymers with short-range interactions, on the lattice or in the continuum.
