We analyze the transport through a narrow ballistic superconductor–normal-metal–superconductor Josephson contact with nonideal transmission at the superconductor–normal-metal interfaces, e.g., due to insulating layers, effective mass steps, or band misfits (SIN interfaces). The electronic spectrum in the normal wire is determined through the combination of Andreev reflection and normal reflection at the SIN interfaces. Strong normal scattering at the SIN interfaces introduces electron- and holelike resonances in the normal region that show up in the quasiparticle spectrum. These resonances have strong implications for the critical supercurrent Ic that we find to be determined by the lowest quasiparticle level: tuning the potential μx0 to the points where electron- and holelike resonances cross, we find sharp peaks in Ic, resulting in a transistor effect. We compare the performance of this resonant Josephson-transistor with that of a superconducting single electron transistor.