© 2017. The American Astronomical Society. All rights reserved. The brief transient emitted as a shock wave erupts through the surface of a pre-supernova star carrying information about the stellar radius and explosion energy. Here, the CASTRO code, which treats radiation transport using multigroup flux-limited diffusion, is used to simulate the light curves and spectra of shock breakout in very low-energy supernovae (VLE SNe), explosions in giant stars with final kinetic energy much less than 1051 erg. VLE SN light curves, computed here with the KEPLER code, are distinctively faint, red, and long-lived, making them challenging to find with transient surveys. The accompanying shock breakouts are brighter, though briefer, and potentially easier to detect. Previous analytic work provides general guidance, but numerical simulations are challenging, due to the range of conditions and lack of equilibration between color and effective temperatures. We consider previous analytic work and extend discussions of color temperature and opacity to the lower energy range explored by these events. Since this is the first application of the CASTRO code to shock breakout, test simulations of normal energy shock breakout of SN 1987A are carried out and compared with the literature. A set of breakout light curves and spectra are then calculated for VLE SNe with final kinetic energies in the range 1047 - 1050erg for red supergiants with main-sequence masses of 15 and 25 M⊙. The importance of uncertainties in stellar atmosphere model, opacity, and ambient medium is discussed, as are observational prospects with current and forthcoming missions.