We investigate a modified one-dimensional contact process with varying infection rates. Specifically, the infection spreads at rate $λ_e$ along the boundaries of the infected region and at rate $λ_i$ elsewhere. We establish the existence of an invariant measure when $λ_i = λ_c$ and $λ_e > λ_c$, where $λ_c$ is the critical parameter of the standard contact process. Moreover, we show that, when viewed from the rightmost infected site, the process converges weakly to this invariant measure. Finally, we prove that along the critical curve within the attractive region of the phase space, the infection almost surely dies out.