This paper is concerned with the numerical solution of the weakly singular Urysohn integral equations. The singularity subtraction technique presented in [P.~M. Anselone, J. Austral. Math. Soc. Ser. B {\bf 22} (1980/81), no.~4, 408--418; MR0626932] is generalized to deal with nonlinear Fredholm integral operators. Two approaches based on this scheme are considered. The first one---the authors call it classical---discretizes the nonlinear problem, and uses a finite-dimensional linearization process to solve numerically the discrete problem. The other approach is based on linearization of the problem in its infinite-dimensional setting, and discretization of the sequence of linear problems by singularity subtraction. The convergence analysis of both approaches is discussed and it is shown that the second approach is more efficient and reliable than the classical one. Three numerical examples are also given which illustrate the theoretical results.