Most existing copula models for dependent censoring in the literature assume that the parameter defining the copula is known. However, prior knowledge on this dependence parameter is often unavailable. In this article we propose a novel model under which the copula parameter does not need to be known. The model is based on a parametric copula model for the relation between the survival time (T) and the censoring time (C), whereas the marginal distributions of T and C follow a semiparametric Cox proportional hazards model and a parametric model, respectively. We show that this model is identified, and propose estimators of the nonparametric cumulative hazard and the finite-dimensional parameters. It is shown that the estimators of the model parameters and the cumulative hazard function are consistent and asymptotically normal. We also investigate the performance of the proposed method using finite-sample simulations. Finally, we apply our model and estimation procedure to a follicular cell lymphoma dataset. Supplementary materials for this article are available online. (with Negera Wakgari Deresa)