  We introduce a constrained priority mechanism that combines outcome-based
matching from machine-learning with preference-based allocation schemes common
in market design. Using real-world data, we illustrate how our mechanism could
be applied to the assignment of refugee families to host country locations, and
kindergarteners to schools. Our mechanism allows a planner to first specify a
threshold $\bar g$ for the minimum acceptable average outcome score that should
be achieved by the assignment. In the refugee matching context, this score
corresponds to the predicted probability of employment, while in the student
assignment context it corresponds to standardized test scores. The mechanism is
a priority mechanism that considers both outcomes and preferences by assigning
agents (refugee families, students) based on their preferences, but subject to
meeting the planner's specified threshold. The mechanism is both strategy-proof
and constrained efficient in that it always generates a matching that is not
Pareto dominated by any other matching that respects the planner's threshold.
