  We prove the filling area conjecture in the hyperelliptic case. In
particular, we establish the conjecture for all genus 1 fillings of the circle,
extending P. Pu's result in genus 0. We translate the problem into a question
about closed ovalless real surfaces. The conjecture then results from a
combination of two ingredients. On the one hand, we exploit integral geometric
comparison with orbifold metrics of constant positive curvature on real
surfaces of even positive genus. Here the singular points are Weierstrass
points. On the other hand, we exploit an analysis of the combinatorics on
unions of closed curves, arising as geodesics of such orbifold metrics.
