  We study an ultracold Bose gas in the presence of 1D disorder for repulsive
inter-atomic interactions varying from zero to the Thomas-Fermi regime. We show
that for weak interactions the Bose gas populates a finite number of localized
single-particle Lifshits states, while for strong interactions a delocalized
disordered Bose-Einstein condensate is formed. We discuss the schematic
quantum-state diagram and derive the equations of state for various regimes.
