Absorbed neutron doses in regions outside the treatment volume from negative pion radiotherapy are presented, based on neutron spectral measurements for pions stopping in a tissue-equivalent target. A Monte Carlo neutron transport computer code was developed and used to calculate the absorbed dose as a function of the distance from the centre of the treatment volume. The Monte Carlo code, which is a modification of a neutron detector efficiency code, follows neutrons and gamma rays as they interact with either hydrogen or oxygen nuclei in a phantom. The code includes neutron elastic scattering on both hydrogen and oxygen as well as five inelastic nuclear reactions on oxygen. The recoil charged particles which provide the absorbed dose are considered until the neutron escapes the phantom or its kinetic energy falls below 1 ke V. Calculations of absorbed dose are compared with earlier dose calculations and measurements. Measurements of the neutron spectrum from a tissue-equivalent target indicate that the total kinetic energy carried away by neutrons is about 76 MeV, which is a significantly higher value than that used in earlier estimates of the neutron dose. The calculations presented here suggest that the neutron dose outside large treatment volumes may limit the use of negative pions for some therapeutic applications.