  We present a new application of the traditional thermodynamic Bethe ansatz to
the spin-1/2 antiferromagnetic uniform Heisenberg chain and derive exact
nonlinear integral equations for just {\em two} functions describing the
elementary excitations. Using this approach the magnetic susceptibility $\chi$
and specific heat C versus temperature T are calculated to high accuracy for
$5\times10^{-25}\leq T/J\leq 5$. The $\chi(T)$ data agree very well at low T
with the asymptotically exact theoretical low-T prediction of S. Lukyanov,
Nucl. Phys. B {\bf 522}, 533 (1998). The unknown coefficients of the second and
third lowest-order logarithmic correction terms in Lukyanov's theory for C(T)
are estimated from the C(T) data.
