  This paper is concerned with the existence and qualitative properties of
positive ground state solutions for the planar Schr\"odinger-Newton equation on
the disc. First, we prove the existence and radial symmetry of all the positive
ground state solutions by employing the symmetric decreasing rearrangement and
Talenti's inequality. Next, we develop Newton's theorem and then use the
contraction mapping principle to establish the uniqueness of the positive
ground state solution for the Schr\"odinger-Newton equation on the disc in the
two dimensional case. Finally, we show that the unique positive ground state
solution converges to the trivial solution as the radius $R$ tending to
infinity, which is totally different from the higher dimensional case in
\cite{Guo-Wang-Yi}.
