This paper explores the geometry and physics of null infinity, $\mathcal{I}^+$, as a weakly isolated horizon (WIH) in the Penrose conformal completion of an asymptotically flat spacetime. The authors show that $\mathcal{I}^+$ naturally inherits the structure of a WIH, even though it is not a submanifold of the physical spacetime. They demonstrate that the differences in the physics associated with $\mathcal{I}^+$ and black hole (or cosmological) horizons $\Delta$ can be traced back to the fact that $\mathcal{I}^+$ is a WIH in the conformal completion rather than the physical spacetime. Specifically, the BMS group at $\mathcal{I}^+$ stems from the symmetry group of WIHs. The paper also discusses the universal structure and symmetry groups of WIHs, showing that the symmetry group of a generic WIH is a 1-dimensional extension of the BMS group. In a companion paper, the authors introduce a new Hamiltonian framework to obtain fluxes and charges associated with symmetries at $\mathcal{I}^+$ and $\Delta$. The results highlight the common mathematical framework underlying $\Delta$ and $\mathcal{I}^+$, paving the way for exploring the relationship between horizon dynamics in the strong field region and waveforms at infinity, as well as the analysis of black hole evaporation in quantum gravity.This paper explores the geometry and physics of null infinity, $\mathcal{I}^+$, as a weakly isolated horizon (WIH) in the Penrose conformal completion of an asymptotically flat spacetime. The authors show that $\mathcal{I}^+$ naturally inherits the structure of a WIH, even though it is not a submanifold of the physical spacetime. They demonstrate that the differences in the physics associated with $\mathcal{I}^+$ and black hole (or cosmological) horizons $\Delta$ can be traced back to the fact that $\mathcal{I}^+$ is a WIH in the conformal completion rather than the physical spacetime. Specifically, the BMS group at $\mathcal{I}^+$ stems from the symmetry group of WIHs. The paper also discusses the universal structure and symmetry groups of WIHs, showing that the symmetry group of a generic WIH is a 1-dimensional extension of the BMS group. In a companion paper, the authors introduce a new Hamiltonian framework to obtain fluxes and charges associated with symmetries at $\mathcal{I}^+$ and $\Delta$. The results highlight the common mathematical framework underlying $\Delta$ and $\mathcal{I}^+$, paving the way for exploring the relationship between horizon dynamics in the strong field region and waveforms at infinity, as well as the analysis of black hole evaporation in quantum gravity.