We realize a strongly coupled double quantum dot in a single all-organic molecule by introducing a nonconjugated bridge in between two identical conjugated moieties. Spin-1/2 Kondo and Kondo enhanced low-energy excitations for, respectively, the odd and even electron occupation are observed in off-resonant transport. The ground state in the even occupation can be the singlet or the triplet state varying between samples. This observation suggests that both antiferromagnetic and ferromagnetic interactions between spins are of the same order of magnitude.