We prove that the group algebra KG of a group G over a field K is primitive, provided that G has a non-abelian free subgroup with the same cardinality as G, and that G satisfies the following condition ( * ): for each subset M of G consisting of a finite number of elements not equal to 1, and for any positive integer m, there exist distinct a, b, and c in G so that if (x −1 1 g 1 x 1 ) · · · (x −1 m g m x m ) = 1, where g i is in M and x i is equal to a, b, or c for all i between 1 and m, then x i = x i+1 for some i. This generalizes results of [1], [9], [18], and [19], and proves that, for every countably infinite group G satisfying ( * ), KG is primitive for any field K. We use this result to determine the primitivity of group algebras of one relator groups with torsion.