Concentrated colloidal alumina dispersions were frozen in a directional solidification apparatus that provides independent control of the freezing rate and temperature gradient. Two distinct steady-state modes of periodic ice banding were observed in the range of freezing rates examined. For each mode, the wavelength between successive bands of segregated ice decreases with increasing freezing rate. At low freezing rates (0.25-3 mu m s(-1)), the ice segregates from the suspension into ice lenses, which are cracklike in appearance, and there is visible structure in the layer of rejected particles in the unfrozen region ahead of the ice lenses. In this regime, we argue that compressive cryosuction forces lead to the irreversible aggregation of the rejected particles into a close-packed cohesive layer. The temperature in the aggregated layer is depressed below the bulk freezing point by more than 2 degrees C before the ice lenses are encountered; moreover, this undercooled region appears as a light-colored layer. The magnitude of the undercooling and the color change in this region both suggest the presence of pore ice and the formation of a frozen fringe. The possibility of a frozen fringe is supported by a quantitative model of the freezing behavior. At intermediate freezing rates, around 4 mu m s(-1), the pattern of ice segregation is disordered, coinciding with the disappearance of the dark and light-colored layers. Finally, at high freezing rates (5-10 mu m s(-1)), there is a new mode of periodic ice banding that is no longer cracklike and is absent of any visible structure in the suspension ahead of the ice bands. We discuss the implications of our experimental findings for theories of ice lensing.