Abstract: We consider integral equations of the first kind, which are associated with the class of ill-posed problems. This class also includes the problem of inversing the integral Laplace transform, which is used to solve a wide class of mathematical problems. Integral equations are reduced to ill-conditioned systems of linear algebraic equations (in which unknowns represent the coefficients of expansion in a series in shifted Legendre polynomials of some function that is simply expressed in terms of the sought original; this function is found as a solution of a certain finite moment problem in a Hilbert space). To obtain a reliable solution of the system, regularization methods are used. The general strategy is to use the Tikhonov stabilizer or its modifications. A specific type of stabilizer in the regularization method is indicated; this type is focused on an a priori low degree of smoothness of the desired original. The results of numerical experiments are presented; they confirm the efficiency of the proposed inversion algorithm.