  Nonanalytic $m_q^{1/2}$ and $m_q\ln m_q$ chiral corrections to the baryon
magnetic moments are computed. The calculation includes contributions from both
intermediate octet and decuplet baryon states. Unlike the one-loop
contributions to the baryon axial currents and masses, the contribution from
decuplet intermediate states does not partially cancel that from octet
intermediate states. The fit to the observed magnetic moments including
$m_q^{1/2}$ corrections is found to be much worse than the tree level SU(3) fit
if values for the baryon-pion axial coupling constants obtained from a tree
level extraction are used. Using the axial coupling constant values extracted
at one loop results in a better fit to the magnetic moments than the tree level
SU(3) fit. There are three linear relations amongst the magnetic moments when
$m_q^{1/2}$ corrections are included, and one relation including $m_q^{1/2}$,
$m_q\ln m_q$ and $m_q$ corrections. These relations are independent of the
axial coupling constants of the baryons and agree well with experiment.
