Description and specifications of a new and distinct blackberry cultivar named ‘A-1960’ which originated from seed produced by a hand pollinated cross of Ark. Selection Ark.-1583 (non-patented, unreleased genotype female)×Ark.-1482 (non-patented, unreleased genotype male) is provided. This new blackberry cultivar can be distinguished by its very firm fruit with long shape, medium-large size, attractive fruit appearance, very good flavor, excellent plant health, and cane erectness.