The electric charge acquired by aqueous droplets when they contact an electrode is a crucial parameter in experimental and industrial applications where electric fields are used to manipulate droplet motion and coalescence. For unclear reasons, many investigators have found that aqueous droplets acquire significantly more positive than negative charge. Extant techniques for determining the droplet charge typically rely on a hydrodynamic force balance that depends on accurate characterization of the drag forces acting on the droplet. Here we present an alternative methodology for measuring the droplet charge via direct measurement of the electric current. As the droplet approaches the electrode the current is observed to gradually increase, followed by a large pulse when the droplet makes apparent contact. We interpret the transient current signals as the superposition of the natural response of an RLC circuit and an induced current described by the Shockley Ramo theory. Nonlinear regression of the observed current to the theoretical model allows for the droplet charge to be extracted, independent of any assumptions about the force balance on the droplet. We demonstrate that regression of the current signal yields charge values that are on average within 4% of charges measured via a force balance. We use the chronocoulometric methodology to investigate how the charge varies with the applied potential, and we demonstrate that deionized water droplets contacting planar electrodes acquire on average 69% more positive charge than negative charge.