The "Auger Muons and Infill for the Ground Array" (AMIGA) project provides direct muon counting capacity to the Pierre Auger Observatory and extends its energy detection range down to 0.3 EeV. It currently consists of 61 detector pairs (a Cherenkov surface detector and a buried muon counter) distributed over a 23.5 km super(2) area on a 750 m triangular grid. Each counter relies on segmented scintillator modules storing a logical train of '0's and '1's on each scintillator segment at a given time slot. Muon counter data is sampled and stored at 320 MHz allowing both the detection of single photoelectrons and the implementation of an offline trigger designed to mitigate multi-pixel PMT crosstalk and dark rate undesired effects. Acquisition is carried out by the digital electronics built around a low power Cyclone III FPGA. This paper presents the digital electronics design, internal and external synchronization schemes, hardware tests, and first results from the Observatory.