We report multi-hour stable trapping and steady state of non-laser-coolable Cs+ ions in a linear Paul trap, achieved via resonant charge-exchange (RCE) cooling with a precisely centered ultracold Cs cloud. Without cold atoms, all ions are lost within 3 minutes. With centered cold atoms, RCE cooling establishes a stable population of ions for 6 hours with no measurable decay – more than two orders of magnitude longer hold times compared to prior hybrid atom-ion systems. We observe a threshold behavior in the steady-state number of ions (Ns): initial ion loadings above Ns ions converge downward to Ns ions and remain constant thereafter; initial loadings below Ns ions show no measurable decay. The dynamics is governed by a competition between ion-ion rf heating and ion-atom collisional cooling. The prolonged, simultaneous ion-atom trapping suggests an upper bound on the three-body recombination-induced Cs2+ formation rate constant: k3 << 1.4 × 10^{-25} cm^6 s^{-1}. This realization of long-lived steady state overcomes a critical limitation of prior hybrid atom-ion systems and enables extended studies of ultracold ion-neutral chemistry, rare inelastic collisions, and sympathetic cooling routes for complex molecular ions.