The triplet s-wave scattering length of Li6 is determined using two-photon photoassociative spectroscopy of the diatomic a3Σ+u state of Li26. The measured binding energy of the highest-lying bound state, combined with knowledge of the potential, determines the s-wave scattering length to be (-2160±250)a0, where a0 is the Bohr radius. This extraordinarily large scattering length signifies a near-threshold resonance. A complete table of singlet and triplet scattering lengths for collisions involving Li6 and Li7 determined from this and our previous spectroscopic investigations is given.