  We investigate weighted Lebesgue space estimates for the Bergman projection
on a simply connected planar domain via the domain's Riemann map. We extend the
bounds which follow from a standard change-of-variable argument in two ways.
First, we provide a regularity condition on the Riemann map, which turns out to
be necessary in the case of uniform domains, in order to obtain the full range
of weighted estimates for the Bergman projection for weights in a
B\'{e}koll\`{e}-Bonami-type class. Second, by slightly strengthening our
condition on the Riemann map, we obtain the weighted weak-type $(1,1)$ estimate
as well. Our proofs draw on techniques from both conformal mapping and dyadic
harmonic analysis.
