  Significant progress has been made to experimentally determine a complete set
of the parity-violating (PV) weak-interaction amplitudes between nucleons. In
this paper we describe the design, construction and operation of the n$^3$He
experiment that was used to measure the PV asymmetry $A_{\mathrm{PV}}$ in the
direction of proton emission in the reaction $\vec{\mathrm{n}} +
{^3}\mathrm{He} \rightarrow {^3}\mathrm{H} + \mathrm{p}$, using the capture of
polarized cold neutrons in an unpolarized gaseous $^3\mathrm{He}$ target. This
asymmetry has was recently calculated \cite{Viviani,Viviani2}, both in the
traditional style meson exchange picture, and in effective field theory (EFT),
including two-pion exchange. The high precision result (published separately)
obtained with the experiment described herein forms an important benchmark for
hadronic PV (HPV) theory in few-body systems, where precise calculations are
possible. To this day, HPV is still one of the most poorly understood aspects
of the electro-weak theory. The calculations estimate the size of the asymmetry
to be in the range of $(-9.4 \rightarrow 3.5)\times 10^{-8}$, depending on the
framework or model. The small size of the asymmetry and the small overall goal
uncertainty of the experiment of $\delta A_{\mathrm{PV}} \simeq 1\times10^{-8}$
places strict requirements on the experiment, especially on the design of the
target-detector chamber. In this paper we describe the experimental setup and
the measurement methodology as well as the detailed design of the chamber,
including results of Garfield++ and Geant4 simulations that form the basis of
the chamber design and analysis. We also show data from commissioning and
production and define the systematic errors that the chamber contributes to the
measured $A_{\mathrm{PV}}$. We give the final uncertainty on the measurement.
