We show that if a closed orientable 2k-manifold K, k > 2, with Euler characteristic χ(K) ≠ -2 admits an exact Lagrangian immersion into C2k with one transverse double point and no other self intersections, then K is diffeomorphic to the sphere. The proof combines Floer homological arguments with a detailed study of moduli spaces of holomorphic disks with boundary in a monotone Lagrangian submanifold obtained by Lagrange surgery on K.