Summary: We introduce a new finite volume scheme for one-dimensional conservation systems. This scheme is based on the local resolution of linearized Riemann problems. Unlike the well-known Roe scheme, we do not prescribe any consistency relation with the conservation system. The numerical flux is also defined by taking the physical flux and the values of the linearized Riemann solution at the interface. This ensures that the scheme is in conservation form and with consistent flux. We present some numerical tests.