  We study a dilute gas of attractive bosons confined in a harmonic cylinder,
i.e. under cylindric confinement due to a transverse harmonic potential. We
introduce a many-body wave function which extends the Bethe ansatz proposed by
McGuire (J. Math. Phys. {\bf 5}, 622 (1964)) by including a variational
transverse Gaussian shape. We investigate the ground state properties of the
system comparing them with the ones of the one-dimensional (1D) attractive Bose
gas. We find that the gas becomes ultra 1D as a consequence of the attractive
interaction: the transverse width of the Bose gas reduces by increasing the
number of particles up to a critical width below which there is the collapse of
the cloud. In addition, we derive a simple analytical expression for the
simmetry-breaking solitonic density profile of the ground-state, which
generalize the one deduced by Calogero and Degasperis (Phys. Rev. A {\bf 11},
265 (1975)). This bright-soliton analytical solution shows near the collapse
small deviations with respect to the 3D mean-field numerical solution. Finally,
we show that our variational Gauss-McGuire theory is always more accurate than
the McGuire theory. In addition, we prove that for small numbers of particles
the Gauss-McGuire theory is more reliable than the mean-field theory described
by the 3D Gross-Pitaevskii equation.
