  We present an analysis of high-resolution ($R \sim 50,000$) infrared K-band
echelle spectra of 14 T Tauri stars in the Orion Nebula Cluster. We model
Zeeman broadening in three magnetically sensitive \ion{Ti}{1} lines near $2.2\
\mu$m and consistently detect kilogauss-level magnetic fields in the stellar
photospheres. The data are consistent in each case with the entire stellar
surface being covered with magnetic fields, suggesting that magnetic pressure
likely dominates over gas pressure in the photospheres of these stars. These
very strong magnetic fields might themselves be responsible for the
underproduction of X-ray emission of T Tauri stars relative to what is expected
based on main-sequence star calibrations. We combine these results with
previous measurements of 14 stars in Taurus and 5 stars in the TW Hydrae
association to study the potential variation of magnetic-field properties
during the first 10 million years of stellar evolution, finding a steady
decline in total magnetic flux with age.
