We propose a test for independence of bivariate censored data under univariate censorship. To do this, we first introduce a process defined by the difference between bivariate survival function estimator proposed by Lin and Ying (1993) and the product of the product-limit estimators (Kaplan and Meier, 1958) for the marginal survival functions, and derive its asymptotic properties under the null hypothesis of independence. We propose a Crame`rvon Mises-type test procedure based on the process. We conduct simulation studies to investigate the finite-sample performance of the proposed test and illustrate the proposed test with a real example.