In this paper, a novel fully-explicit weakly compressible solver is developed for solving incompressible two-phase flows. The two-phase flow is modelled by coupling the general pressure equation, momentum conservation equations and the conservative level set advection equation. A HLLC-type Riemann solver is proposed to evaluate the convective fluxes along with a simple, consistent and oscillation-free discretization for the non-conservative terms. The solver is tested against several two-phase flow problems for its robustness and adaptability on structured as well as unstructured meshes.