Inexpensive superconducting wires with low AC losses would open up for a large superconductor market in AC electrical power applications. One candidate for this market is the MgB2 conductor. In the development of an AC optimized superconductor, high-quality measurements of the AC losses under application-like conditions must be available. This article describes an apparatus built for this purpose. The measurement method is calorimetric. The temperature increase of the superconductor sample is measured and compared to the temperature increase due to a heater with known heat input. The system is designed for measurements at temperatures down to 20 K, and for magnetic field strengths up to 1 T (peak) combined with transport currents up to several hundred amperes. Results from tests verifying the capabilities of the system are given, as well as from initial AC loss measurements on a tape-shaped MgB2 superconductor.