Precise values for the critical threshold for the three-dimensional "Swiss cheese" continuum percolation model have been calculated using extensive Monte Carlo simulations. These simulations used a growth algorithm and memory blocking scheme similar to what we used previously in three-dimensional lattice percolation. The simulations yield a value for the critical number density n(c) = 0.652 960 +/-0.000 005, which confirms recent work but extends the precision by two significant figures. (C) 2001 American Institute of Physics.