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Abstract: We consider the one-constant Landau-de Gennes (LdG) model for nematic liquid crystals with traceless tensor field Q as the order parameter that seeks to minimize a Dirichlet energy plus a double well potential that confines the eigenvalues of Q (examples/applications will be described). Moreover, we constrain Q to be uniaxial, which involves a rank-1 constraint. Building on similarities with the one-constant Ericksen energy, we propose a structure-preserving finite element method for the computation of equilibrium configurations. We prove stability and consistency of the method without regularization, and $\Gamma$-convergence of the discrete energies towards the continuous one as the mesh size goes to zero. We also give a monotone gradient flow scheme to find minimizers. We illustrate the method's capabilities with several numerical simulations in two and three dimensions including non-orientable line fields. In addition, we do a direct comparison between the standard LdG model, and the uniaxially constrained model.