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We study equilibrium properties of a cold two-component Fermi gas confined ina quasi-one-dimensional trap of the transverse size $l_{\perp}$. In the dilutelimit ($nl_{\perp}\ll 1$, where $n$ is the 1D density) the problem is exactlysolvable for an arbitrary 3D fermionic scattering length $a_{F}$. When$l_{\perp}/a_{F}$ goes from $-\infty$ to $+\infty$, the system successivelypasses three regimes: weakly interacting Fermi gas, hard core Bose gas andweakly coupled Bose gas. The regimes are separated by two crossovers at$a_{F}\sim \pm nl_{\perp}^{2}$. In conclusion we discuss experimentalimplications of these results.Comment: RevTex4, 4 pages, 2 eps figures; revised version with new references  and extended discussion of the physics of qusi-1D "BCS-BEC" crossove

Dilute Fermi Gas in Quasi-One-Dimensional Traps: From Weakly Interacting Fermions via Hard Core Bosons to a Weakly Interacting Bose Gas