Thank you, Vitaliy. It seems simple, but I could not find the function in the docs. Of course, it is there, now that I know what it is called. : )
By the way, your example has a stray line that I eliminated by reordering the points in the graphic:
pic = Graphics[{EdgeForm[{Thick, Black}],
Red,
FilledCurve[{BezierCurve[{{3, 0}, {2, 1.5}, {-.5, 2.25}, {0, 6}}],
Line[{{0, 6}, {3, 6}}], Line[{{3, 6}, {3, 0}}]}],
Blue,
FilledCurve[{BezierCurve[{{3, 0}, {4, 1.5}, {6.5, 2.25}, {6, 6}}],
Line[{{6, 6}, {3, 6}}], Line[{{3, 6}, {3, 0}}]}]}];
Thanks again!