A theory of nuclear magnetic resonance line shapes in dense paramagnetic insulators at T= is developed by the use of the diagrammatic method for high-temperature two-point spin-correlation functions developed earlier. This theory contains no ad hoc assumptions about the line shapes and reduces to Moriya's theory if only the electron autocorrelation function is kept and if a Gaussian form is assumed for that function. In particular, the equations for the nuclear dipole spin-spin correlation functions are obtained, and it is shown that the expected Lorentzian behavior of the spectral functions follows directly from the lowest-order solutions to these equations without any prior suppositions about line shapes. These spectral functions are proportional to the NMR line shapes in dense paramagnetic insulators. The NMR linewidths which result from this calculation are shown to agree well with experiment in several different nuclei in several different substances.