Fluid product dispenser having a reservoir (R) with a threaded neck (R1) forming an external helical thread (R2), a closure element (T) having a threaded skirt (F1) forming an internal helical thread (F2), and a band (H) engaging around the threaded skirt (F1). The skirt has tabs (F11), each forming a segment (F21) of the internal helical thread (F2). The fastening member (F) has an angular indexing mechanism for positioning it with respect to the external helical thread (R2) with a determined angular orientation, and the segments (F21) of the internal helical thread (F2) of the threaded skirt (F1) come into corresponding threaded contact with the external helical thread (R2) of the neck (R1) in the final assembly position.