  We propose a variational approximation to the ground state energy of a
one-dimensional gas of interacting bosons on the continuum based on the Bethe
Ansatz ground state wavefunction of the Lieb-Liniger model. We apply our
variational approximation to a gas of dipolar bosons in the single mode
approximation and obtain its ground state energy per unit length. This allows
for the calculation of the Tomonaga-Luttinger exponent as a function of density
and the determination of the structure factor at small momenta. Moreover, in
the case of attractive dipolar interaction, an instability is predicted at a
critical density, which could be accessed in lanthanide atoms.
