Dipole excited integrated horn antennas etched in Si/GaAs are a very promising class of antennas for millimeter and submillimeter applications. Areas of application include imaging and tracking through rain and fog, remote sensing, radio astronomy, and plasma diagnostics. In this paper (Part I), the far-field pattern and input impedance of a dipole-fed horn antenna in a ground plane are calculated using full-wave analysis. The solution is based on the numerical evaluation of the pertinent Green's function for the horn structure and the application of the method of moments. The convergence characteristics of the full-wave analysis method are investigated, along with the resonant properties of the strip-dipole and the corresponding behavior of the far-field patterns. The theoretical results are compared in Part II with microwave and millimeter-wave measurements.