A propulsion unit having a propeller for an aircraft comprising: - a nacelle which is intended to be attached to a structural element of the aircraft ; - a propeller which is rotatably mounted in the nacelle about a longitudinal axis of rotation (X) by means of a hub , the propeller comprising blades each of which is mounted pivotally integral with an associated blade cuff which is pivotally mounted about a radial pitch axis (Y) with respect to the hub , and each blade being pivotally mounted with respect to the blade cuff about a folding axis (Z) orthogonal to the radial pitch axis (Y); - a folding device which comprises a folding actuator controlling the pivoting of each blade relative to its blade cuff between a deployed position in which the blades extend radially with respect to the axis (X) of rotation and a folded position in which the blades extend generally longitudinally against the nacelle ; the folding device comprising a transmission associated with each blade , each transmission comprising: - a movable control member which is mounted integral in rotation with the blade cuff and which is moved by the folding actuator , and - a link comprising a first end pivotally mounted on a root of the associated blade eccentrically with respect to the folding axis (Z) and a second end pivotally mounted on the movable control member , characterised in that the control member of each transmission is formed by a slider which is mounted so as to be radially slidable in the associated blade cuff between an internal extreme position corresponding to one of the positions of the associated blade and an external extreme position corresponding to the other of the positions of the associated blade . - a nacelle which is intended to be attached to a structural element of the aircraft ; - a propeller which is rotatably mounted in the nacelle about a longitudinal axis of rotation (X) by means of a hub , the propeller comprising blades each of which is mounted pivotally integral with an associated blade cuff which is pivotally mounted about a radial pitch axis (Y) with respect to the hub , and each blade being pivotally mounted with respect to the blade cuff about a folding axis (Z) orthogonal to the radial pitch axis (Y); - a folding device which comprises a folding actuator controlling the pivoting of each blade relative to its blade cuff between a deployed position in which the blades extend radially with respect to the axis (X) of rotation and a folded position in which the blades extend generally longitudinally against the nacelle ; the folding device comprising a transmission associated with each blade , each transmission comprising: - a movable control member which is mounted integral in rotation with the blade cuff and which is moved by the folding actuator , and - a link comprising a first end pivotally mounted on a root of the associated blade eccentrically with respect to the folding axis (Z) and a second end pivotally mounted on the movable control member , - a movable control member which is mounted integral in rotation with the blade cuff and which is moved by the folding actuator , and - a link comprising a first end pivotally mounted on a root of the associated blade eccentrically with respect to the folding axis (Z) and a second end pivotally mounted on the movable control member , characterised in that the control member of each transmission is formed by a slider which is mounted so as to be radially slidable in the associated blade cuff between an internal extreme position corresponding to one of the positions of the associated blade and an external extreme position corresponding to the other of the positions of the associated blade . <EOS>