A precision calorimeter was used to measure the energy deposited in a liquid-helium-cooled copper target as a result of scattered nuclear radiation from an accelerator target in order to simulate the thermal response of a superconducting magnet under such conditions. The methodology selected for the accelerator application was to measure precisely the temperature rise of a copper target in a vacuum, and to convert the results into energy deposition using the specific heat. Therefore, precise measurements of the specific heat of the copper target were necessary to achieve the required precision and accuracy in the energy deposition measurements for the accelerator application. A pulse-heating technique to measure the specific heat was developed in which electrical square waves were delivered to a surface-mounted heater on the copper target, and the temperature rise of the copper target was measured. The specific heat of the copper was measured from 6 to 8 K and compared to data in the literature. The data were also used to infer the electronic- and lattice-specific-heat coefficients and the Debye temperature for the copper that was used. Comparisons of the data to the specific heat and specific-heat coefficients in the literature are presented. The present technique is shown to be useful for the measurement of the specific heat of good thermal conductors, especially metals at cryogenic temperatures, when very high precision is required.