The IceCube Upgrade will augment the existing IceCube Neutrino Observatory by deploying 700 additional optical sensor modules and calibration devices within its center at a depth of 1.5 to 2.5 km in the Antarctic ice. One goal of the Upgrade is to improve the positioning calibration of the optical sensors to increase the angular resolution for neutrino directional reconstruction. An acoustic calibration system will be deployed to explore the capability of achieving this using trilateration of propagation times of acoustic signals. Ten Acoustic Modules (AM) capable of sending and receiving acoustic signals with frequencies from 5 to 30 kHz will be installed within the detector volume. Additionally, compact acoustic sensors inside 15 optical sensor modules will complement the acoustic calibration system. With this system, we aim for an accuracy of a few tens of cm to localize the Acoustic Modules and sensors. Due to the longer attenuation length of sound compared to light within the ice, acoustic position calibration is especially interesting for the upcoming IceCube-Gen2 detector, which will have a string spacing of around 240 m. In this contribution we present an overview of the technical design of the Acoustic Module as well as results of performance tests with a first complete prototype.