We present a first-principles theoretical approach to calculate the lattice thermal conductivity of diamond based on an exact solution of the Boltzmann transport equation. Density-functional perturbation theory is employed to generate the harmonic and third-order anharmonic interatomic force constants that are required as input. A central feature of this approach is that it provides accurate representations of the interatomic forces and at the same time introduced no adjustable parameters. The calculated lattice thermal conductivities for isotopically enriched and naturally occurring diamond are both in very good agreement with experimental data. The role of the scattering of heat-carrying acoustic phonons by optic branch phonons is also investigated. We show that inclusion of this scattering channel is indispensable in properly describing the thermal conductivity of semiconductors and insulators. The accurate adjustable-parameter-free results obtained herein highlight the promise of this approach in providing predictive descriptions of the lattice thermal conductivity of materials.