We study the effective elastic behaviour of the incompatibly prestrained thin plates, characterized by a Riemann metric G on the reference configuration. We assume that the prestrain is " weak " , i.e. it induces scaling of the incompatible elastic energy E^h of order less than h^2 in terms of the plate's thickness h. We essentially prove two results. First, we establish the Γ-limit of the scaled energies h^{−4} E^h and show that it consists of a von Kármán-like energy, given in terms of the first order infinitesimal isometries and of the admissible strains on the surface isometrically immersing G_{2×2} (i.e. the prestrain metric on the midplate) in R^3. Second, we prove that in the scaling regime E^h~ h^β with β > 2, there is no other limiting theory: if inf h^{−2} E^h → 0 then inf E^h ≤ Ch^4 , and if inf h^{−4} E^h → 0 then G is realizable and hence min E^h = 0 for every h.