  We have used the XAFS (x-ray-absorption fine structure) technique to
investigate the local structure about the Cu, Ba, and Tl atoms in orthorhombic
Tl-2201 with a superconducting transition temperature T$_c$=60 K. Our results
clearly show that the O(1), O(2), Cu, and Ba atoms are at their ideal sites as
given by the diffraction measurements, while the Tl and O(3) atoms are more
disordered than suggested by the average crystal structure. The Tl-Tl distance
at 3.5 \AA{ } between the TlO layers does not change, but the Tl-Tl distance at
3.9 \AA{ } within the TlO layer is not observed and the Tl-Ba and Ba-Tl peaks
are very broad. The shorter Tl-O(3) distance in the TlO layer is about 2.33
\AA, significantly shorter than the distance calculated with both the Tl and
O(3) atoms at their ideal $4e$ sites ( $x=y=$0 or $\frac{1}{2}$). A model based
on these results shows that the Tl atom is displaced along the $<110>$
directions from its ideal site by about 0.11 \AA; the displacements of
neighboring Tl atoms are correlated. The O(3) atom is shifted from the $4e$
site by about 0.53 \AA{ } roughly along the $<100>$ directions. A comparison of
the Tl L$_{III}$-edge XAFS spectra from three samples, with T$_c$=60 K, 76 K,
and 89 K, shows that the O environment around the Tl atom is sensitive to T$_c$
while the Tl local displacement is insensitive to T$_c$ and the structural
symmetry. These conclusions are compared with other experimental results and
the implications for charge transfer and superconductivity are discussed. This
paper has been submitted to Phys. Rev. B.
