Excited states of the neutron-rich nuclei 97,99Rb were populated for the first time using the multistep Coulomb excitation of radioactive beams. Comparisons of the results with particle-rotor model calculations provide clear identification for the ground-state rotational band of 97Rb as being built on the πg9/2 [431] 3/2+ Nilsson-model configuration. The ground-state excitation spectra of the Rb isotopes show a marked distinction between single-particle-like structures below N=60 and rotational bands above. The present study defines the limits of the deformed region around A∼100 and indicates that the deformation of 97Rb is essentially the same as that observed well inside the deformed region. It further highlights the power of the Coulomb-excitation technique for obtaining spectroscopic information far from stability. The 99Rb case demonstrates the challenges of studies with very short-lived postaccelerated radioactive beams.