  We present high resolution (R ~ 60,000) circular spectropolarimetry of the
classical T Tauri star TW Hydrae. We analyze 12 photospheric absorption lines
and measure the net longitudinal magnetic field for 6 consecutive nights. While
no net polarization is detected the first five nights, a significant
photospheric field of Bz = 149 \pm 33 G is found on the sixth night. To rule
out spurious instrumental polarization, we apply the same analysis technique to
several non-magnetic telluric lines, detecting no significant polarization. We
further demonstrate the reality of this field detection by showing that the
splitting between right and left polarized components in these 12 photospheric
lines shows a linear trend with Lande g-factor times wavelength squared, as
predicted by the Zeeman effect. However, this longitudinal field detection is
still much lower than that which would result if a pure dipole magnetic
geometry is responsible for the mean magnetic field strength of 2.6 kG
previously reported for TW Hya. We also detect strong circular polarization in
the He I 5876 and the Ca II 8498 emission lines, indicating a strong field in
the line formation region of these features. The polarization of the Ca II line
is substantially weaker than that of the He I line, which we interpret as due
to a larger contribution to the Ca II line from chromospheric emission in which
the polarization signals cancel. However, the presence of polarization in the
Ca II line indicates that accretion shocks on Classical T Tauri stars do
produce narrow emission features in the infrared triplet lines of Calcium.
