The n = 2 fine-structure transitions in positronium have been measured
by observation of increased Lyman-alpha radiation at microwave induce
d resonances. We used an accelerator based slow positron beam to produ
ce positronium in the 2S-3(1) excited state inside a waveguide. Our re
sults are nu0 = 18499.65 +/-1.20+/-4.00 MHz, nu1 = 13012.42+/-0.67+/-1
.54 MHz, and nu2 = 8624.38+/-0.54+/-1.40 MHz for the 2S-3(1) --> 2P-3(
0,1,2) transition frequencies, respectively. The first error is statis
tical and the second is systematic. This is in agreement with recent b
ound state QED calculations.