We present ab initio calculations of the internal C-C bond dissociation curve for single molecules of (cis-1,4) polyisoprene and polybutadiene. We define "bond rupture" as that point on the reaction coordinate where the unrestricted Kohn-Sham, or diradical, solution falls below the restricted, or closed-shell, solution. Using this definition, we find that rupture occurs at a tensile force of 6.8 nN for polyisoprene and 7.2 nN for polybutadiene. Their respective rupture strains are 45% and 42%. Our calculations show that the energy density versus extension is not sensitive to the number of isoprene units contained in the molecule, i.e., it is essentially independent of the chain length. These relatively large rupture strains have important implications for understanding the failure mechanism in rubber, and imply that purely enthalpic chain stretching must commence well before tensile failure occurs.