  The bulk and boundary magnetizations are calculated for the critical Ising
model on a randomly triangulated disk in the presence of a boundary magnetic
field h. In the continuum limit this model corresponds to a c = 1/2 conformal
field theory coupled to 2D quantum gravity, with a boundary term breaking
conformal invariance. It is found that as h increases, the average
magnetization of a bulk spin decreases, an effect that is explained in terms of
fluctuations of the geometry. By introducing an $h$-dependent rescaling factor,
the disk partition function and bulk magnetization can be expressed as
functions of an effective boundary length and bulk area with no further
dependence on h, except that the bulk magnetization is discontinuous and
vanishes at h = 0. These results suggest that just as in flat space, the
boundary field generates a renormalization group flow towards h = \infty. An
exact analytic expression for the boundary magnetization as a function of $h$
is linear near h = 0, leading to a finite nonzero magnetic susceptibility at
the critical temperature.
