This paper develops a likelihood-based inference procedure for continuous-time capture-recapture models. The first-capture and recapture intensities are assumed to be in constant proportion but may otherwise vary arbitrarily through time. The full likelihood is partitioned into two factors, one of which is analogous to the likelihood in a special type of multiplicative intensity model arising in failure time analysis. The remaining factor is free of the non-parametric nuisance parameter and is easily maximized. This factor provides an estimator of population size and an asymptotic variance under a counting process framework. The resulting estimation procedure is shown to be equivalent to that derived from a martingale-based estimating function approach. Simulation results are presented to examine the performance of the proposed estimators.