The mechanism includes a display member (10) fitted with a toothing (14) mounted so as to move in rotation, a drive wheel set (39) mounted so as to move in rotation, including a cam (40) and a finger (24), wherein the finger is secured in rotation to the cam to make a finger-cam assembly and the finger cooperate with the toothing to drive the display member by steps, and a spring (38) cooperating at least indirectly with the cam, wherein the cam is arranged for pivoting, the finger-cam assembly, via the slackening action of the spring, so as to drive the display member through one step. The cam is further arranged for pivoting the finger assembly, via the slackening action of the spring, so as to release the finger from the toothing after the finger cam assembly has driven the display member through one step.