In many industrial applications inversion of the Laplace transform is necessary to reconstruct a searched for object function. In X-ray diffractometry, a method from non-destructive testing, we want to recover the stress tensor of a specimen with the help of X-ray measurements. The mathematical model is described by the Laplace transform of the stress tensor at finitely many scanning points that are not equally distributed. In this article we present an inversion method for the Laplace transform using a non-equispaced sampling grid that applies the approximate inverse to this transform. The approximate inverse is a regularization technique for inverse problems based on evaluations of the given data with so called reconstruction kernels. Each kernel solves a system of linear equations defined by the adjoint of the Laplace transform and dilation invariant mollifiers, which are designed particularly for this operator. An error estimate is given and point-wise convergence of the approximate inverse to the exact solution is proved. The paper ends with some numerical results.