A new type of low-regularity integrator is proposed for the Navier-Stokes equations. Unlike the other low-regularity integrators for nonlinear dispersive equations, which are all fully explicit in time, the proposed method is a semi-implicit exponential method in time in order to preserve the energy-decay structure of the Navier-Stokes equations. First-order convergence of the proposed method is established independently of the viscosity coefficient μ, under weaker regularity conditions than other existing numerical methods, including the semi-implicit Euler method and classical exponential integrators. The proposed low-regularity integrator can be extended to full discretization with either a stabilized finite element method or a spectral collocation method in space, as illustrated in this article. Numerical results show that the proposed method is much more accurate than the semi-implicit Euler method in the viscous case μ=O(1), and more stable than the classical exponential integrator in the inviscid case μ→0.