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We consider the following Kirchhoff type problem involving a critical nonlinearity: ⎧ $-[a+b({\int }_{\Omega }{\left|\nabla u\right|²dx)}^m{]\Delta u=f(x,u)+|u|}^{2*-2}u$ in Ω, ⎨ ⎩ u = 0 on ∂Ω, where $\Omega \subset {ℝ}^N$ (N ≥ 3) is a smooth bounded domain with smooth boundary ∂Ω, a > 0, b ≥ 0, and 0 < m < 2/(N-2). Under appropriate assumptions on f, we show the existence of a positive ground state solution via the variational method.