  The electric dipole moment (EDM) enhancement factor of atomic Tl is of
considerable interest as it has been used in determining the most accurate
intrinsic electron EDM limit to date. However, the value of this quantity
varies from $-179$ to $-1041$ in different approximations. In view of the large
uncertainties associated with many of these calculations, we have employed the
relativistic coupled-cluster theory with single and double excitations and a
subset of leading triple excitations and obtained the EDM enhancement factor of
Tl as $-466$, which in combination with the most accurate measured value of Tl
EDM yields $2.0 \times 10^{-27}\,{\mathrm e\,cm}$ as the new upper limit for
the EDM of the electron. The importance of all-order correlation effects is
emphasized and their trends are compared with those of two other ab initio
calculations.
