  We study the magnetism of a layered, spin-$\frac12$ organic-inorganic copper
sulfate, which is a close realization of the star lattice antiferromagnet, one
of the playgrounds of geometric frustration and resonating valence bond physics
in two spatial dimensions. Our thermodynamic measurements show no ordering down
to 0.1 K and a characteristic field-induced entropic shift, revealing the
presence of an infinite number of competing states down to very low energy
scales. The response to external magnetic fields shows, in addition, a peculiar
anisotropy, reflected in the formation of a 1/3 magnetization plateau (stable
up to full saturation around 105 T) and a paramagnetic, Curie-like
susceptibility for one direction of the field (${\bf H}\parallel{\bf c}$), and
a completely different response in other field directions. Our first-principles
density functional theory calculations and exact diagonalizations show that
these experimental puzzles are distinctive signatures of a strong interplay
between geometric frustration and sizable Dzyaloshinskii-Moriya interactions,
and the emergence of a continuous U(1) symmetry at low energy scales.
