  Partial disorder --the microscopic coexistence of long-range magnetic order
and disorder-- is a rare phenomenon, that has been experimental and
theoretically reported in some Ising- or easy plane-spin systems, driven by
entropic effects at finite temperatures. Here, we present an analytical and
numerical analysis of the $S=1/2$ Heisenberg antiferromagnet on the
$\sqrt{3}\times \sqrt{3}$-distorted triangular lattice, which shows that its
quantum ground state has partial disorder in the weakly frustrated regime. This
state has a 180$^\circ$ N\'eel ordered honeycomb subsystem, coexisting with
disordered spins at the hexagon center sites. These central spins are
ferromagnetically aligned at short distances, as a consequence of a
Casimir-like effect originated by the zero-point quantum fluctuations of the
honeycomb lattice.
