  In this paper, we provide new proofs of the existence and the condensation of
Bethe roots for the Bethe Ansatz equation associated with the six-vertex model
with periodic boundary conditions and an arbitrary density of up arrows (per
line) in the regime $\Delta<1$. As an application, we provide a short, fully
rigorous computation of the free energy of the six-vertex model on the torus,
as well as an asymptotic expansion of the six-vertex partition functions when
the density of up arrows approaches $1/2$. This latter result is at the base of
a number of recent results, in particular the rigorous proof of
continuity/discontinuity of the phase transition of the random-cluster model,
the localization/delocalization behaviour of the six-vertex height function
when $a=b=1$ and $c\ge1$, and the rotational invariance of the six-vertex model
and the Fortuin-Kasteleyn percolation.
