In this paper we present a new quadrature method for computing Galerkin stiffness matrices arising from the discretisation of 3D boundary integral equations using continuous piecewise linear boundary elements. This rule takes as points some subset of the nodes of the mesh and can be used for computing non-singular Galerkin integrals corresponding to pairs of basis functions with non-intersecting supports. When this new rule is combined with standard methods for the singular Galerkin integrals we obtain a `hybrid' Galerkin method which has the same stability and asymptotic convergence properties as the true Galerkin method but a complexity more akin to that of a collocation or Nyström method. The method can be applied to a wide range of singular and weakly-singular first- and second-kind equations, including many for which the classical Nyström method is not even defined. The results apply to equations on piecewise-smooth Lipschitz boundaries, and to non-quasiuniform (but shape-regular) meshes. A by-product of the analysis is a stability theory for quadrature rules of precision 1 and 2 based on arbitrary points in the plane. Numerical experiments demonstrate that the new method realises the performance expected from the theory.