A thermal system in the presence of a weakly confining potential never relaxes to equilibrium, since the Boltzmann-Gibbs distribution is non-normalized. Using a modified entropy maximization principle, in the spirit of the canonical ensemble from standard thermodynamics, we show that a non-normalizable Boltzmann infinite-density replaces in this case the standard Boltzmann distribution (the latter applies in the presence of strongly-binding potentials), and it allows us to obtain both the time- and the ensemble-average of any observable integrable with respect to it (with surprising consequences!), as well as the entropy-energy relation in the system and a generalized virial theorem. By merging infinite-ergodic theory with Boltzmann-Gibbs statistics, we extend the tool-box of thermodynamics beyond its traditional limits, while shedding new light on the concept of ergodicity.