  We observed the classical T Tauri star TW Hya with \textit{HST}/STIS using
the E140M grating, from 1150--1700 \AA, with the E230M grating, from 2200--2900
\AA, and with \FUSE from 900--1180 \AA. Emission in 143 Lyman-band H$_2$ lines
representing 19 progressions dominates the spectral region from 1250--1650 \AA.
The total H$_2$ emission line flux is $1.94 \times 10^{-12}$ erg cm$^{-2}$
s$^{-1}$, which corresponds to $1.90\times10^{-4}$ $L_\odot$ at TW Hya's
distance of 56 pc. A broad stellar \Lya line photoexcites the H$_2$ from
excited rovibrational levels of the ground electronic state to excited
electronic states. The \ion{C}{2} 1335 \AA doublet, \ion{C}{3} 1175 \AA\
multiplet, and \ion{C}{4} 1550 \AA doublet also electronically excite H$_2$.
The velocity shift of the H$_2$ lines is consistent with the photospheric
radial velocity of TW Hya, and the emission is not spatially extended beyond
the $0\farcs05$ resolution of \textit{HST}. The H$_2$ lines have an intrinsic
FWHM of $11.91\pm0.16$ \kms. One H$_2$ line is significantly weaker than
predicted by this model because of \ion{C}{2} wind absorption. We also do not
observe any H$_2$ absorption against the stellar \Lya profile. From these
results, we conclude that the H$_2$ emission is more consistent with an origin
in a disk rather than in an outflow or circumstellar shell. We also analyze the
hot accretion-region lines (e.g., \ion{C}{4}, \ion{Si}{4}, \ion{O}{6}) of TW
Hya, which are formed at the accretion shock, and discuss some reasons why Si
lines appear significantly weaker than other TR region lines.
