We present a new numerical code that solves the general relativistic magneto-hydrodynamical (GRMHD) equations coupled to the Einstein equations for the evolution of a dynamical spacetime within a conformally-flat approximation. This code has been developed with the main objective of studying astrophysical scenarios in which both, high magnetic fields and strong gravitational fields appear, such as the magneto-rotational collapse of stellar cores, the collapsar model of GRBs, and the evolution of neutron stars. The code is based on an existing and thoroughly tested purely hydrodynamical code and on its extension to accommodate weakly magnetized fluids (passive magnetic-field approximation). These codes have been applied in the past to simulate the aforementioned scenarios with increasing levels of sophistication in the input physics. The numerical code we present here is based on high-resolution shock- capturing schemes to solve the GRMHD equations, which are cast in first-order, flux-conservative hyperbolic form, together with the flux constraint transport method to ensure the solenoidal condition of the magnetic field. Since the astrophysical applications envisaged do not deviate significantly from spherical symmetry, the conformal flatness condition approximation is used for the formulation of the Einstein equations; this has repeatedly shown to yield very good agreement with full general relativistic simulations of core- collapse supernovae and the evolution of isolated neutron stars. In addition, the code can handle several equations of state, from simple analytical expressions to microphysical tabulated ones. In this paper we present stringent tests of our new GRMHD numerical code, which show its ability to handle all aspects appearing in the astrophysical scenarios for which the code is intended, namely relativistic shocks, highly magnetized fluids, and equilibrium configurations of magnetized neutron stars. As an application, magneto- rotational core-collapse simulations of a realistic progenitor are presented and the results compared with our previous findings in the passive magnetic-field approximation.