We consider the viscous Burgers equation on the real line with additive white-in-time random force and study its ergodic properties. In the case of space-homogeneous noise, under mild assuptions we establish the existence of a unique ergodic solution for every fixed average velocity. The analysis utilizes the directed polymer models derived from the Feynman--Kac representation of the Burgers solution, where the question of unique ergodicity can be reformulated in terms of the existence and uniqueness of the infinite-volume polymer Gibbs measures. Furthermore, the inviscid limit holds for these ergodic solutions, and correspondingly, the polymer measures in the zero-temperature limit converges to delta measures concentrated on infinite minimizers. Further questions on the finer properties of this global solution are open problems related to the KPZ universality conjectures for non-integrable models. In the case of L2 noise with small damping, we present a polynomial mixing result using Foiaş-Prodi type estimates, which holds for general conservation laws as well.