  We study equilibrium properties of a cold two-component Fermi gas confined in
a quasi-one-dimensional trap of the transverse size $l_{\perp}$. In the dilute
limit ($nl_{\perp}\ll 1$, where $n$ is the 1D density) the problem is exactly
solvable for an arbitrary 3D fermionic scattering length $a_{F}$. When
$l_{\perp}/a_{F}$ goes from $-\infty$ to $+\infty$, the system successively
passes three regimes: weakly interacting Fermi gas, hard core Bose gas and
weakly coupled Bose gas. The regimes are separated by two crossovers at
$a_{F}\sim \pm nl_{\perp}^{2}$. In conclusion we discuss experimental
implications of these results.
