Thirty rovibrational transitions of H-3(+) have been observed near 1.25 mu m using a tunable diode laser and a positive column discharge. In addition to the H-3(+) transitions, over 200 transitions between Rydberg states of H-2 were observed-these could be discriminated against by using a discharge dominated by He, which apparently collisionally quenches the Rydberg states of H-2 without affecting H-3(+). Twenty-eight of the H-3(+) transitions have been assigned to the nu(1) + 2 nu(2)(2)<-- 0 band, and provide experimentally determined energy levels for most of the levels up to J = 4 in the nu(1) + 2 nu(2)(2) state. The remaining two H-3(+) transitions have been assigned to the 2 nu(1) + nu(2)<-- 0 band. These bands represent a crucial test of ab initio calculations, as they reach higher vibrational levels of H-3(+) than any yet observed. We have compared our experimental results with recent variational calculations by several groups.