In this paper, we propose a new simplified lattice Boltzmann method (SLBM) for magnetohydrodynamic flows that outperforms the classical one in terms of accuracy, while preserving its advantages. A very recent paper [A. De Rosis, J. Al-Adham, H. Al-Ali and R. Meng, "Double-D2Q9 lattice Boltzmann models with extended equilibrium for two-dimensional magnetohydrodynamic flows", Phys. Fluids 33, 035143 (2021)] demonstrated that the SLBM enforces the divergence-free condition of the magnetic fiield in an excellent manner and involves the lowest amount of virtual memory. However, the SLBM is characterised by the poorest accuracy. Here, the two-stages algorithm that is typical of the SLBM is replaced by a one-stage procedure following the approach devised for non-conductive fluids in a very recent effort [A. Delgado-Gutierrez, P. Marzocca, D. Cardenas, and O. Probst, "A single-step and simplified graphics processing unit lattice Boltzmann method for high turbulent flows", International Journal for Numerical Methods in Fluids ( 2021)]. The Chapman-Enskog expansion formally demonstrates the consistency of the present scheme. The resultant algorithm is very compact and easy to be implemented. Given all these features, we believe that the proposed approach is an excellent candidate to perform numerical simulations of two-and three-dimensional magnetohydrodynamic flows.