  Two-dimensional electron gases (2DEGs) with high mobility, engineered in
semiconductor heterostructures host a variety of ordered phases arising from
strong correlations, which emerge at sufficiently low temperatures. The 2DEG
can be further controlled by surface gates to create quasi-one dimensional
systems, with potential spintronic applications. Here we address the
long-standing challenge of cooling such electrons to below 1$\,$mK, potentially
important for identification of topological phases and spin correlated states.
The 2DEG device was immersed in liquid $^3$He, cooled by the nuclear adiabatic
demagnetization of copper. The temperature of the 2D electrons was inferred
from the electronic noise in a gold wire, connected to the 2DEG by a metallic
ohmic contact. With effective screening and filtering, we demonstrate a
temperature of 0.9$\,\pm\,$0.1$\,$mK, with scope for significant further
improvement. This platform is a key technological step, paving the way to
observing new quantum phenomena, and developing new generations of
nanoelectronic devices exploiting correlated electron states.
