  Recent experimental and theoretical work has shown that there are conditions
in which a trapped, low-density Bose gas behaves like the one-dimensional
delta-function Bose gas solved years ago by Lieb and Liniger. This is an
intrinsically quantum-mechanical phenomenon because it is not necessary to have
a trap width that is the size of an atom -- as might have been supposed -- but
it suffices merely to have a trap width such that the energy gap for motion in
the transverse direction is large compared to the energy associated with the
motion along the trap. Up to now the theoretical arguments have been based on
variational - perturbative ideas or numerical investigations. In contrast, this
paper gives a rigorous proof of the one-dimensional behavior as far as the
ground state energy and particle density are concerned. There are four
parameters involved: the particle number, $N$, transverse and longitudinal
dimensions of the trap, $r$ and $L$, and the scattering length $a$ of the
interaction potential. Our main result is that if $r/L\to 0$ and $N\to\infty$
the ground state energy and density can be obtained by minimizing a
one-dimensional density functional involving the Lieb-Liniger energy density
with coupling constant $\sim a/r^2$.
