A drive apparatus in a self-propelled construction machine comprises a milling rotor and a main drive which can be coupled to the milling rotor via an engaging and disengaging clutch, and an auxiliary drive which can be coupled to the milling rotor alternatively to the main drive. The clutch is provided with a fixed side and an axially displaceable output side, which is displaceable between an engaged position and a disengaged position. The axially displaceable side is simultaneously also the engaging and disengaging part of a shiftable transmission, the fixed part of which is driven by the auxiliary drive, with the transmission and the clutch being engaged and disengaged in a selective fashion via the axially displaceable side of the clutch.