Natural radiocarbon is a useful tracer for evaluating deep-ocean circulation rates in models. However, it takes several thousand years for the deep-ocean concentration of natural radiocarbon to come to equilibrium with surface fluxes making it computationally too expensive to routinely simulate it with moderate- to high-resolution ocean models. To overcome this difficultly we have developed an implicit solver for computing prebomb radiocarbon that requires the equivalent of only a few tens of model years to reach equilibrium. The solver uses a Newton–Krylov algorithm with a preconditioner based on a coarse-grained annually-averaged tracer-transport operator. We have implemented the solver for the ocean component of the Community Earth System Model (CESM), and used it to simulate the prebomb equilibrium radiocarbon distribution.  The model resolution (1x1 deg in the horizontal and 60 levels in the vertical) is the highest of all natural radiocarbon simulations performed to date. By comparing the modeled radiocarbon distribution to observations we were able to identify clear biases in the circulation of the deep Pacific Ocean. The biases with prebomb radiocarbon ages that are twice the observed values are significantly larger than those of coarser resolution models for which radiocarbon was used to tune sub-grid-scale parameterizations.  Our new solver will make it possible to take advantage of natural radiocarbon observations when calibrating sub-grid-scale parameterizations in models of increasing resolution.