The stability of an exponential current in water to infinitesimal perturbations in the presence of gravity and capillarity is investigated. Some new results on the generation of gravitycapillary waves are presented which supplement the previous works of Morland, Saffman & Yuen (1991) and Young & Wolfe (2014), namely in finite depth. To consider perturbations of much larger scales, a specific attention is paid to the stability of the exponential current only in the presence of gravity.