Provided is an apparatus for driving a boom of a construction machine, allowing required power to be reduced. The apparatus includes a boom cylinder, a variable-displacement hydraulic pump , a control valve guiding hydraulic oil discharged by the hydraulic pump to the boom cylinder, a boom-raising-operation detector boom-cylinder pressure detectors, a supply selector valve switchable between a permitting position for permitting the hydraulic oil to be supplied from the hydraulic pump a head-side chamber of the boom cylinder and a blocking position for blocking the supply, a supply oil passage permitting hydraulic oil to be supplied to the head-side chamber during the blocking, and a controller which brings the supply selector valve into the blocking position and reduces volume of the hydraulic pump upon judging that excavation reaction force extends the boom cylinder even with no supply of hydraulic oil from the hydraulic pump to the head-side chamber.