Transverse-energy distributions have been measured in the pseudorapidity region -0.1<ηlab<2.9 for oxygen-nucleus collisions at incident energies of 60 and 200 GeV per nucleon for Al, Ag, and W target nuclei. The cross-section for the heaviest target nuclei at the highest incident energy is measured over 5 orders of magnitude and out to a maximum ET of 200 GeV. Measurements of the differential density dET/dηlab as a function of ηlab are presented. In the pseudorapidity region -0.1<ηlab<2.9 the transverse energy for an average central collision is proportional to ≈A0.5, while the fraction of the total transverse energy measured in an extended region of ηlab>2.9 decreases with increasing A. The distributions are compared with the predictions of a dual parton model. Finally the question of the energy density is addressed.