[en] We study the effect of an additional simple BCS model interaction in a system in which the dominant scattering causes marginal-Fermi-liquid (MFL) behavior. It leads to a superconductor that shows a number of unusual features as compared to weak-coupling BCS superconductors. The superconducting transition temperature is lowered due to the MFL pair-breaking effects. If the bosonic mode that causes the MFL behavior, and which presumably is of electronic origin, is largely suppressed below T_c due to the gap opening up, Δ(0) remains practically unaffected by the MFL behavior. As a result, the ratio 2Δ(0)/k_BTc is enhanced, and the gap opens up much more rapidly than in the standard BCS approximation; even a gap discontinuity at T_c is possible. Furthermore, we find a suppression of the coherence peak in the nuclear-spin-relaxation rate and a more dramatic decrease below T_c of the ultrasonic attenuation rate and the penetration depth