We study the large scale fluctuations of the KPZ equation in dimensions d \geq 3 driven by Gaussian noise that is white in time Gaussian but features non-integrable spatial correlation with decay rate \kappa \in (2, d) and a suitable limiting profile. We show that its scaling limit is described by the corresponding additive stochastic heat equation. In contrast to the case of compactly supported covariance, the noise in the stochastic heat equation retains spatial correlation with covariance |x|^{-\kappa}. Surprisingly, the noise driving the limiting equation turns out to be the scaling limit of the noise driving the KPZ equation so that, under a suitable coupling, one has convergence in probability, unlike in the case of integrable correlations where fluctuations are enhanced in the limit and convergence is necessarily weak.
