  The QCD Axion is a particle postulated to exist since the 1970s to explain
the Strong-CP problem in particle physics. It could also account for all of the
observed Dark Matter in the universe. The Axion Resonant InterAction DetectioN
Experiment (ARIADNE) experiment intends to detect the QCD axion by sensing the
fictitious "magnetic field" created by its coupling to spin. The experiment
must be sensitive to magnetic fields below the $10^{-19}$ T level to achieve
its design sensitivity, necessitating tight control of the experiment's
magnetic environment. We describe a method for controlling three aspects of
that environment which would otherwise limit the experimental sensitivity.
Firstly, a system of superconducting magnetic shielding is described to screen
ordinary magnetic noise from the sample volume at the $10^8$ level. Secondly, a
method for reducing magnetic field gradients within the sample up to $10^2$
times is described, using a simple and cost-effective design geometry. Thirdly,
a novel coil design is introduced which allows the generation of fields similar
to those produced by Helmholtz coils in regions directly abutting
superconducting boundaries. The methods may be generally useful for magnetic
field control near superconducting boundaries in other experiments where
similar considerations apply.
