Long-time asymptotic behavior for the viscous Burgers equation on the real line is considered. When there is a non-negative and compactly supported Radon measure as a stationary source, we prove that solutions of the viscous Burgers equation converge to a positive, bounded, and nondecreasing steady state by finding an almost optimal convergence order. The non-integrability of the steady state only allows local convergence on compact subsets, hence a Véron-type argument must be modified by adopting a proper weight function.