A new relativistic valence-bond method including configuration interaction is suggested for calculations of the electronic structure of diatomic molecules. The basis set of one-electron orbitals includes relativistic Hartree-Fock atom-type orbitals obtained by numerically solving the Hartree-Fock-Dirac equations for free atoms-ions or atoms in an external field. The basis set also includes virtual atomic states described by the Hartree-Fock-Sturm relativistic orbitals. The complete wave function is obtained as a linear combination of Slater determinants constructed from localized atomic-type orbitals nonorthogonal on different centers. Alongside covalent configurations, ionic charge-transfer configurations are included in the calculations. In the one-configuration variant, the method is equivalent to the relativistic Heitler-London model. The suggested method is designed for calculating molecules with heavy atoms; it is an all-electron method, which does not require the use of pseudopotentials or effective core potentials for excluding core states. The use of the numerical Hartree-Fock basis set is especially effective in calculations of such physical characteristics as densities on the nucleus, superfine splitting parameters, X-ray transition energies, chemical shifts, etc. The calculation data on the spectral characteristics of the AgH and AgH⁺ molecules are given. The relativistic calculation results are compared with the data of nonrelativistic calculations performed with the use of the same basis set and in the same configuration space by a similar nonrelativistic method.