We study graviton-graviton scattering in partial-wave amplitudes after unitarizing their Born terms. In order to apply S-matrix techniques, based on unitarity and analyticity, we introduce an S-matrix associated to this resummation that is free of infrared divergences. This is achieved by removing the diverging phase factor calculated by Weinberg that multiplies the S matrix, and that stems from the virtual infrared gravitons. A scalar graviton-graviton resonance with vacuum quantum numbers (JPC =0++) is obtained as a pole in the nonperturbative S-wave amplitude, which we call the graviball. Its resonant effects along the physical real-s axis may peak at values substantially lower than the UV cutoff squared of the theory. For some scenarios, this phenomenon could have phenomenological consequences at relatively low-energy scales, similarly to the σ resonance in QCD.