The need for providing reliable numerical methods for the solution of weakly singular Volterra integral equations of first kind stems from the fact that they are connected to important problems in the theory and applications of stochastic processes. In the first section the above problems and some peculiarities of such equations are briefly sketched. Section 2 describes a method for obtaining an approximate solution whose properties are described in section 3: such properties guarantee that our approximate solution always oscillates around the rigorous one. Section 4 discusses the applicability of our case of some classical bounds on the errors. The remaining sections are all devoted to the construction of upper bounds on the oscillating error in order to reach an high degree of reliability for our solution. All the bounds are independent of the numerical method which is employed for obtaining the numerical solution.