Highly accurate relativistic Gaussian basis sets with a finite-nucleus model are developed for the 103 elements from H (Z=1) to Lr (Z=103). The present GTO sets augment the relativistic basis sets with a point-charge model proposed in the first paper of this series. The relativistic third-order Douglas-Kroll approach is adopted in optimizing the orbital exponents of a basis set by minimizing the atomic self-consistent field (SCF) energy. The basis sets are designed to have equal quality and to be appropriate for the incorporation of relativistic effects. The performance of the present basis sets is tested by calculations on a prototypical molecule, gold dimer using SCF and the singles and doubles coupled-cluster model with perturbative triples [CCSD(T)]. Several spectroscopic constants are calculated for the ground state of Au-2. At the basis set superposition error (BSSE) corrected CCSD(T) level, the deviation from experiment is DeltaR(e)=0.018 Angstrom, Deltaomega(e)=-3 cm(-1), and DeltaD(e)=-0.17 eV. The finite-size nucleus effect makes R-e, omega(e), and D-e smaller by 0.004 parallel to, 1 cm(-1), and 0.05 eV, respectively. The application shows that the present relativistic Gaussian-type orbitals (GTO) basis sets with a finite-nucleus model are accurate and reliable.