A mechanical infusion pump device for injecting medication into a patient's IV, includes an IV reservoir and a syringe communicated to the IV reservoir and having a syringe plunger connected to a filling plunger that is moved with the syringe plunger in a first syringe-filling direction to fill the syringe with fluid medication. Biasing weight(s) is/are provided on the filling plunger for exerting bias or force from gravity on the filling plunger and the syringe plunger in a second syringe-discharging direction to dispense medication from the syringe into the patient's IV when the filling plunger is released. Only a calibrated amount of medicine can be discharged to the patent's IV over time as determined by calibration of the biasing weights and the metering element located between the syringe and the patient's IV for a given viscosity of the medication.