Iceberg calving accounts for around half of the ice lost annually from Antarctica, but realistic representation of fracture and calving in large-scale ice sheet models remains a major unsolved problem in glaciology. We present a new phase-field viscoelastic model for fracture that simulates the slow deformation of ice and the distribution and evolution of cracks. Cracks nucleate and propagate in response to the evolving stress field, and are influenced by water pressure below sea level. The model incorporates nonlinear-viscous rheology, linear-elastic rheology, and a phase-field variational formulation, which allows simulation of complex fracture phenomena. We show that this approach is capable of simulating the physical process of calving. Numerical experiments supported by a simplified model suggest that calving rate will scale with the fourth power of ice thickness for a floating ice front that has no variation across flow. The equations make no assumptions about the style of calving, so they would also simulate numerous more realistic settings in Antarctica for which material parameters and three-dimensional effects can be expected to influence the calving rate.