  A new non-perturbative method to compute accurate oscillation modes in
rapidly rotating stars is presented. In this paper, the effect of the
centrifugal force is fully taken into account while the Coriolis force is
neglected. This assumption is valid when the time scale of the oscillation is
much shorter than the inverse of the rotation rate and is expected to be
suitable for high radial order p-modes of $\delta$ Scuti stars. Axisymmetric
p-modes have been computed in uniformly rotating polytropic models of stars. In
the frequency and rotation range considered, we found that as rotation
increases (i) the asymptotic structure of the non-rotating frequency spectrum
is first destroyed then replaced by a new form of organization (ii) the mode
amplitude tends to concentrate near the equator (iii) differences with
perturbative methods become significant as soon as the rotation rate exceeds
about fifteen percent of the Keplerian limit. The implications for the
seismology of rapidly rotating stars are then discussed.
