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For the group O(p, q) we give a new construction of its minimal unitary represen-tation via Euclidean Fourier analysis. This is an extension of the q = 2 case, where the representation is the mass zero, spin zero representation realized in a Hilbert space of solutions to the wave equation. The group O(p, q) acts as the Möbius group of conformal transformations on Rp−1,q−1, and preserves a space of solutions of the ultrahyperbolic Laplace equation on Rp−1,q−1. We construct in an intrinsic and natural way a Hilbert space of solutions so that O(p, q) becomes a continuous irreducible unitary representation in this Hilbert space. We also prove that this rep-resentation is unitarily equivalent to the representation on L2(C), where C is the conical subvariety of the nilradical of a maximal parabolic subalgebra obtained by intersecting with the minimal nilpotent orbit in the Lie algebra of O(p, q). Content