  By an extension of the Bethe ansatz method used by Gwa and Spohn, we obtain
an exact expression for the large deviation function of the time averaged
current for the fully asymmetric exclusion process in a ring containing $N$
sites and $p$ particles. Using this expression we easily recover the exact
diffusion constant obtained earlier and calculate as well some higher
cumulants. The distribution of the deviation $y$ of the average current is, in
the limit $N \to \infty$, skew and decays like $\exp - (A y^{5/2})$ for $y \to
+ \infty$ and $\exp - (A' |y|^{3/2})$ for $y \to -\infty$. Surprisingly, the
large deviation function has an expression very similar to the pressure (as a
function of the density) of an ideal Bose or Fermi gas in $3d$.
