•A new lattice Boltzmann model for the one-fluid multiphase flow equations.•Surface tension is set precisely a priori and independent of other parameters.•Narrow interfaces are captured accurately and efficiently.•Reduced need for on-grid differencing.•Excellent agreement with theoretical and benchmark results. A multiphase lattice Boltzmann model is constructed to numerically solve the one-fluid flow equations for immiscible fluids. The method features one solver for the macroscopic pressure and momentum and another for a scalar field that captures and sharpens the interface. The surface tension is set a priori and independently of other parameters. The interface capillary tensor is embedded within the moments of the lattice Boltzmann equation so that its divergence is captured locally. The algorithm is simple and can compute flows with large density and viscosity ratios while maintaining distributed but narrow interfaces. The model is validated against analytical solutions and benchmark simulations.