A pump for a fluid product, the pump comprising a frame, a piston, and a drive system for driving the piston to translate the piston relative to the frame back and forth along a displacement axis (A-A'), the drive system comprising an input shaft extending along a rotation axis (B-B'), a motor adapted to rotate said input shaft about the axis of rotation (B-B') relative to the frame, and a conversion system for converting the rotation of the input shaft about the axis of rotation (B-B') into translational motion of the piston, the conversion system comprising a rack and pinion reciprocating system, the rack and pinion reciprocating system comprising at least one half-pinion kinematically linked to the input shaft so that rotation of the input shaft causes rotation of the or each half-pinion, a first rack integral with the piston and which is able to engage the half-pinion or a first of the half-pinions, in a first direction, and a second rack integral with the piston and which is able to engage the half-pinion or a second of the half-pinions, in a second direction opposite the first direction, the conversion system comprising a carriage mounted for translational movement relative to the frame, the piston being integral with said carriage and the racks (94, 96) forming part of said carriage, the rack and pinion reciprocating system having at least one disengaged configuration in which no half-pinion is meshed with either the first rack or the second rack, and the conversion system comprises a mechanism for immobilising the piston when the rack and pinion reciprocating system is in the disengaged configuration, said immobilising mechanism comprising: - at least one stop surface (112, 114, 116, 118) carried by the carriage, and - an eccentric kinematically linked to the input shaft such that rotation of the input shaft causes rotation of said eccentric about an eccentric axis, the eccentric being adapted to bear against the abutment surface (112, 114, 116, 118) or at least one of the abutment surfaces (112, 114, 116, 118) when the rack and pinion reciprocating system is in a disengaged configuration, characterised in that the eccentric axis is formed by the axis of rotation (B-B'). the rack and pinion reciprocating system comprising at least one half-pinion kinematically linked to the input shaft so that rotation of the input shaft causes rotation of the or each half-pinion, a first rack integral with the piston and which is able to engage the half-pinion or a first of the half-pinions, in a first direction, and a second rack integral with the piston and which is able to engage the half-pinion or a second of the half-pinions, in a second direction opposite the first direction, the conversion system comprising a carriage mounted for translational movement relative to the frame, the piston being integral with said carriage and the racks (94, 96) forming part of said carriage, the rack and pinion reciprocating system having at least one disengaged configuration in which no half-pinion is meshed with either the first rack or the second rack, and the conversion system comprises a mechanism for immobilising the piston when the rack and pinion reciprocating system is in the disengaged configuration, said immobilising mechanism comprising: - at least one stop surface (112, 114, 116, 118) carried by the carriage, and - an eccentric kinematically linked to the input shaft such that rotation of the input shaft causes rotation of said eccentric about an eccentric axis, the eccentric being adapted to bear against the abutment surface (112, 114, 116, 118) or at least one of the abutment surfaces (112, 114, 116, 118) when the rack and pinion reciprocating system is in a disengaged configuration, - at least one stop surface (112, 114, 116, 118) carried by the carriage, and - an eccentric kinematically linked to the input shaft such that rotation of the input shaft causes rotation of said eccentric about an eccentric axis, the eccentric being adapted to bear against the abutment surface (112, 114, 116, 118) or at least one of the abutment surfaces (112, 114, 116, 118) when the rack and pinion reciprocating system is in a disengaged configuration, characterised in that the eccentric axis is formed by the axis of rotation (B-B').