  We study the low-temperature critical behavior of the one-dimensional Hubbard
model near half filling caused by enhanced antiferromagnetic fluctuations. We
use a mean-field-type approximation with a two-particle self-consistency
renormalizing the bare interaction. It allows us to control a transition from
high to low temperatures as well as from weak to strong-coupling. We show that
there is a crossover temperature $T_{0}= t\exp\{-1/U\rho(0)\}$ for arbitrary
interaction $U>0$ and the bare density of states at the Fermi energy
$\rho(0)>0$. The solution at lower temperatures goes over to strong coupling
and approaches a quantum critical point with the diverging staggered
susceptibility and a gap in the excitation spectrum at zero temperature.
