The 2^3P_1-to-2^3P_2 fine-structure interval in atomic helium is measured using the frequency-offset separated-oscillatory-fields (FOSOF) technique. Two temporally separated microwave fields set up excitation paths that accumulate different quantum-mechanical phases. To detect the atoms that have changed states due to the microwaves, these atoms are excited to a Rydberg state and Stark ionized. The number of resulting ions is counted on a channel electron multiplier. In a typical SOF experiment, the relative phase between the two microwave pulses is toggled between 0 and 180, and the change in the signal amplitude between the two phases is detected as a function of applied microwave frequency. In the FOSOF technique, two microwave pulses with a slight frequency offset are applied to the atoms. The relative phase seen by the atoms changes continuously due to the frequency offset, leading to a sinusoidally oscillating atomic signal. The phase of the oscillating signal is measured with respect to the phase of a reference generated by combining the frequency-offset microwaves. The phase difference between the oscillating atomic signal and reference signal crosses zero at resonance and changes linearly as a function of applied microwave frequency. Major signal-to-noise ratio (SNR) enhancement has been achieved by employing a two-dimensional magneto-optical trap and by using Stark-ionization detection. The excellent SNR allows for a very extensive study of systematic effects. A wide range of experiment parameters has been investigated. The final measured result is 2 291 176 590(25) Hz. This is the most precise measurement of the interval to date and thus the most precise test of the two-electron quantum-electrodynamics theory. When the 2^3P_0-to-2^3P_1 transition is measured at the same level of precision and the combined result of the 2^3P_0-to-2^3P_2 fine-structure interval is compared with a sufficiently precise theory, a sub-part-per-billion determination of the fine-structure constant using a two-electron system will become possible for the first time. Comparison with other fine-structure constant measurements could lead to tests of possible beyond-the-Standard-Model physics.