In this extended abstract, a surprising connection is described between a specific brand of random lattices, namely planar quadrangulations, and Aldous' Integrated SuperBrownian Excursion (ISE). As a consequence, the radius r_n of a random quadrangulation with n faces is shown to converge, up to scaling, to the width r=R-L of the support of the one-dimensional ISE The combinatorial ingredients are an encoding by well labelled trees, reminiscent of the work of Cori and Vauquelin, and the conjugation of tree principle, used to relate the latter trees to embedded (discrete) plane trees in the sense of Aldous. {From} probability, we need a new result of independent interest, namely the weak convergence of the encoding of a random embedded plane tree by two contour walks (e^{(n)},\hat W^{(n)}) to the Brownian snake description (e,\hat W) of ISE.