  It has been proposed that the pseudogap state of underdoped cuprate
superconductors may be due to a transition to a phase which has circulating
currents within each unit cell. Here, we use polarized neutron diffraction to
search for the corresponding orbital moments in two samples of underdoped
YBa$_2$Cu$_3$O$_{6+x}$ with doping levels $p=0.104$ and 0.123. In contrast to
some other reports using polarized neutrons, but in agreement with nuclear
magnetic resonance and muon spin rotation measurements, we find no evidence for
the appearance of magnetic order below 300 K. Thus, our experiment suggests
that such order is not an intrinsic property of high-quality cuprate
superconductor single crystals. Our results provide an upper bound for a
possible orbital loop moment which depends on the pattern of currents within
the unit cell. For example, for the CC-$\theta_{II}$ pattern proposed by Varma,
we find that the ordered moment per current loop is less than 0.013 $\mu_B$ for
$p=0.104$.
