We have carried out a major survey for visual binaries toward the Orion Nebula Cluster, using images obtained with the Advanced Camera for Surveys on the Hubble Space Telescope through an Hα filter. From 1051 stars more than 60'' from θ¹ Ori C, we have selected 781 that fulfill the criteria for membership in the Orion Nebula Cluster. Among these, we find 78 multiple systems (75 binaries and 3 triples), of which 55 are new discoveries, in the range from 0.1'' to 1.5''. We perform a statistical study of the 72 binaries and 3 triples that have separations in the limited range 0.15'' to 1.5'', within which we need no incompleteness correction. An analysis of the stellar density in our images suggests that of these binaries, nine are line-of-sight associations. When corrected for this, we find a binary fraction of 8.8% ± 1.1% within the limited separation range from 67.5 to 675 AU (counting the three triples as six binaries). The field binary fraction in the same range from Duquennoy & Mayor is a factor of 1.5 higher. Within the range 150-675 AU that overlaps the study of binaries in T Tauri associations by Reipurth & Zinnecker, we find that the associations have a factor of 2.2 more binaries than the Orion Nebula Cluster, in approximate agreement with earlier results based on data from the inner Trapezium region with small-number statistics. The binary separation distribution function of the Orion Nebula Cluster shows unusual structure, with a sudden steep decrease in the number of binaries as the separation increases beyond 0.5'', corresponding to 225 AU. We have measured the ratio of binaries wider than 0.5'' to binaries closer than 0.5'' as a function of distance from the Trapezium, and we find that this ratio is significantly depressed in the inner region of the Orion Nebula Cluster. The deficit of binaries with larger separations in the central part of the cluster is likely due to dissolution or orbital change of the wider binaries during their passage through the potential well of the inner cluster region. All of our primaries appear to be T Tauri stars, with the exception of one Herbig Ae/Be star, and there are indications that a substantial number of secondaries could be brown dwarfs.