  We create rapidly rotating Bose-Einstein condensates in the lowest Landau
level, by spinning up the condensates to rotation rates $\Omega>99%$ of the
centrifugal limit for a harmonically trapped gas, while reducing the number of
atoms. As a consequence, the chemical potential drops below the cyclotron
energy $2\hbar\Omega$. While in this mean-field quantum Hall regime we still
observe an ordered vortex lattice, its elastic shear strength is strongly
reduced, as evidenced by the observed very low frequency of Tkachenko modes.
Furthermore, the gas approaches the quasi-two-dimensional limit. The associated
cross-over from interacting- to ideal-gas behavior along the rotation axis
results in a shift of the axial breathing mode frequency.
