  We show that under general conditions there is at least one natural
inflationary direction for the Kahler moduli of type IIB flux
compactifications. This requires a Calabi-Yau which has h^{2,1}>h^{1,1}>2 and
for which the structure of the scalar potential is as in the recently found
exponentially large volume compactifications. We also need - although these
conditions may be relaxed - at least one Kahler modulus whose only
non-vanishing triple-intersection is with itself and which appears by itself in
the non-perturbative superpotential. Slow-roll inflation then occurs without a
fine tuning of parameters, evading the eta problem of F-term inflation. In
order to obtain COBE-normalised density perturbations, the stabilised volume of
the Calabi-Yau must be O(10^5-10^7) in string units, and the inflationary scale
M_{infl} ~ 10^{13} GeV. We find a robust model independent prediction for the
spectral index of 1 - 2/N_e = 0.960 - 0.967, depending on the number of
efoldings.
