The Gas Amplifier Detector with Germanium Tagging (GADGET) is a new detection system devoted to the measurement of weak, low-energy $beta$-delayed proton decays relevant for nuclear astrophysics studies. It is comprised of a new gaseous Proton Detector equipped with a Micromegas readout for charged particle detection, surrounded by the existing Segmented Germanium Array (SeGA) for the high-resolution detection of the prompt $gamma$-rays. In this work we describe in detail for the first time the design, construction, and operation of the GADGET system, including performance of the Proton Detector. We present the results of a recent commissioning experiment performed with textsuperscript{25}Si beam at the National Superconducting Cyclotron Laboratory (NSCL). GADGET provided low-background, low-energy $beta$-delayed proton detection with efficiency above 95%, and relatively good efficiency for proton-gamma coincidences (2.7% at 1.37 MeV).
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