A new kernel is proposed for the accurate simulation of planar antennas printed on ultra-thin substrates by means of integral-equation methods. A scalar-potential representation is used and a number of spectral-domain asymptotic terms, in the form of a series of exponentials, are extracted and integrated into the spatial domain. The spatial-domain counterpart of the remainder is evaluated numerically via Hankel transforms to be simply convoluted numerically with basis and testing functions. So, a global Method of Moments (MoM) impedance matrix is obtained from the asymptotic and remaining parts, which allows for an efficient evaluation of the electromagnetic properties of antennas printed on very thin substrates. It is shown that the asymptotic parts can be estimated in closed form, which is validated against purely numerical or hybrid (numerical and analytical) solutions.