A study is carried out for one-dimensional Frenkel excitons near the band center in a system with nearest-neighbor off-diagonal random interactions. The uncorrelated and the correlated (positional disorder) cases are considered for chains of 10(7) atoms. The low-energy spectrum and its localization properties are investigated through the integrated density of states and the inverse localization length. We have tested the predictions of the scaling theory developed for the uncorrelated case against our numerical data and found good agreement as the energy approaches the band center. We have also established that the correlated case, with a general, realistic random interaction also shows scaling behavior with an amplitude constant that is twice as large as in the uncorrelated case. [S0163-1829(99)04630-5].