In 1984, Camina and Gagen gave the result that the block-transitive automorphism group G of a 2-v k ( , , 1)design with k v must be flag-transitive, moreover, G is point-primitive of affine or almost simple type. As a generalization of this result, the purpose of this paper is to study block-transitive automorphism groups of nontrivial 2-v k λ ( , , ) designs with r k ( , ) = 1, where r is the number of blocks incident with a given point. We prove