We apply a DC-electric field model to the analysis of soft and hard X-ray observations of a solar flare observed by Yohkoh and the Compton Gamma Ray Observatory (CGRO) on 6 September 1992. The flare was observed simultaneously in the soft X-ray Ca XIX line by the Yohkoh Bragg Crystal Spectrometer (BCS) and in hard X-rays (greater than 50 keV) by the CGRO Burst and Transient Spectrometer Experiment (BATSE). A strong stationary component of Ca XIX emission was present at the start of impulsive hard X-ray emission indicating an extended phase of heating prior to the production of energetic nonthermal electrons. We interpret the preflare Ca XIX emission as a signature of Joule heating by field-aligned currents. We relate the temporal variation of impulsive hard X-ray emission to the rate of runaway electron acceleration by the DC-electric field associated with the current. We find that the initial rise in hard X-ray emission is consistent with electron acceleration by a DC-electric field that increased from a preflare value of less than approximately 10(exp -5) V/cm to approximately (9 +/- 1) x 10(exp -5) V/cm at the time of the first hard X-ray peak and then remained constant during the rest of the impulsive phase. We attribute the increase in electric field strength to the formation of a current sheet at the reconnection point of two loop structures. The decrease in hard X-ray emission after flare maximum is consistent with a reduction in the number of runaway electrons due to an increase in coronal density produced by chromospheric evaporation. The increased density quenches the runaway process by enhancing collisional thermalization of electrons. To avoid the generation of an unrealistically large magnetic field, the flaring region must be highly filamented into greater than approximately 10(exp 6) oppositely directed current channels of approximately 30 cm width with an initial preflare current of approximately 3 x 10(exp 10) A per channel.