  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).
