A weak force of attraction between electrons and thermal neutrons is indicated in recent experiments by Havens, Rabi, and Rainwater, and by Fermi and Marshall. According to meson theory this would be expected, since the neutron is considered as one charge state of a nucleon (the proton is the other state) which is coupled to a meson field. The attraction is then interpreted as an electrostatic scattering of the neutron which exists part of the time as a proton and meson. We perform a third order perturbation calculation in the approximation of weak coupling between mesons and nucleons. Neutrons and protons are here treated as Dirac particles which are coupled to a meson field of spin zero. The results indicate that the observed interaction is suitably described in terms of the meson field.