We present in this paper an algorithm to solve pure-convection problems with a conservative Lagrange-Galerkin formulation in the framework of the finite element method. The integrals obtained from the Lagrange-Galerkin formulation will be computed with an algorithm which leads to conservation of mass up to machine accuracy, when we transfer information from the mesh moved by the characteristic curves of the convection operator to the current mesh. The algorithm to compute the integrals considers the intersection of meshes composed by triangles (2-dimensions) and tetrahedra (3-dimensions) with straight sides. We will illustrate the good features of the method in terms of stability, accuracy and mass conservations in different pure-convection tests with non-divergence-free velocity fields.