The surface detector array of the Pierre Auger Observatory consists of 1660 water-Cherenkov detector stations (WCDs) that sample the charged particles and photons of air showers initiated by energetic cosmic rays. Each station continuously samples signals from photomultiplier tubes (PMTs) viewing the water volume, and records a approximately 20 micro-second trace whenever a local trigger condition is satisfied. Absolute timing is provided by a GPS receiver on each station, and power is provided by a solar power system. The Observatory is currently implementing an upgrade, "AugerPrime". AugerPrime includes the addition of a small PMT to increase dynamic range, a plastic scintillator above each WCD to enhance the separation of electromagnetic and muon shower components, a radio detector to measure the radio emission of inclined air showers, and an infill of buried muon counters to provide additional cross checks. Consequently, new electronics to support these additional detectors is also being implemented. In addition to more measurement channels, the new electronics includes improved GPS receivers, higher sampling frequency, increased dynamic range, increased processing capability, and improved calibration and monitoring systems. This paper presents the design of the new electronics and discusses performance characteristics observed in laboratory measurements and engineering array data.