  We condition super-Brownian motion on "boundary statistics" of the exit
measure $X_D$ from a bounded domain $D$. These are random variables defined on
an auxiliary probability space generated by sampling from the exit measure
$X_D$. Two particular examples are: conditioning on a Poisson random measure
with intensity $\beta X_D$ and conditioning on $X_D$ itself. We find the
conditional laws as $h$-transforms of the original SBM law using Dynkin's
formulation of $X$-harmonic functions. We give explicit expression for the
(extended) $X$-harmonic functions considered. We also obtain explicit
constructions of these conditional laws in terms of branching particle systems.
For example, we give a fragmentation system description of the law of SBM
conditioned on $X_D=\nu$, in terms of a particle system, called the backbone.
Each particle in the backbone is labeled by a measure $\tilde{\nu}$,
representing its descendants' total contribution to the exit measure. The
particle's spatial motion is an $h$-transform of Brownian motion, where $h$
depends on $\tilde{\nu}$. At the particle's death two new particles are born,
and $\tilde{\nu}$ is passed to the newborns by fragmentation.
