Positronium is the quasistable bound system consisting of an electron and its antiparticle, the positron. Its energy levels can be explained to a high degree of accuracy by the electromagnetic interaction, affording an ideal test of the quantum electrodynamic (QED) theory of bound systems. We have measured the 1 3 S 1–2 3 S 1 interval in positronium by Doppler-free two-photon spectroscopy to be 1 233 607 216.4±3.2 MHz. We employ continous-wave (cw) excitation to eliminate the problems inherent with pulsed laser measurements of nonlinear transitions. Positronium (Ps) atoms generated in vacuum are excited to the 2S state using cw laser light built up to 2.5 kW circulating power in a resonant Fabry-Pérot cavity. The excited-state atoms are photoionized using a pulsed laser at 532 nm, and the liberated positrons counted as the cw laser is tuned relative to a reference line in tellurium (Te 2) molecular vapor. The fit of a detailed theoretical model to the measured line shape determines the Ps resonance frequency relative to the Te 2 reference line. The Monte Carlo model includes details of the excitation and detection geometry, the Ps velocity distribution, and the dynamic Stark shift, and gives excellent agreement with the measured line shapes. The quoted 2.6 parts per billion (ppb) uncertainty is dominated by the measurement of the Ps line center relative to the Te 2 reference line, with a 1.0-ppb contribution from a recent calibration of our Te 2 cell relative to the hydrogen 1S-2S transition frequency. The measurement is in excellent agreement with theory and sufficiently accurate to provide a test of the as-yet-uncalculated α 4 R∞ QED correction. Our measurement tests the α 2 R∞ QED contributions to the energy of the 1 3 S 1 and 2 3 S 1 states to 3.5 parts in 10 5.