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The aim of this paper is to prove the following result. Consider the criticalIsing model on the rescaled grid $a\mathbb{Z}^2$, then the renormalizedmagnetization field \[\Phi^a:=a^{15/8}\sum_{x\ina\mathbb{Z}^2}\sigma_x\delta_x,\] seen as a random distribution (i.e.,generalized function) on the plane, has a unique scaling limit as the mesh size$a\searrow0$. The limiting field is conformally covariant.

Planar Ising magnetization field I. Uniqueness of the critical scaling limit