We report the discovery of two low-mass companions to the young A0V star HD 1160 at projected separations of 81 + or - 5 AU (HD 1160 B) and 533 + or - 25 AU (HD 1160 C) by the Gemini NICI Planet-Finding Campaign. Very Large Telescope images of the system taken over a decade for the purpose of using HD 1160 A as a photometric calibrator confirm that both companions are physically associated. By comparing the system to members of young moving groups and open clusters with well-established ages, we estimate an age of 50 super(+50) sub(-40) Myr for HD 1160 ABC. While the UVW motion of the system does not match any known moving group, the small magnitude of the space velocity is consistent with youth. Near-IR spectroscopy shows HD 1160 C to be an M3.5 + or - 0.5 star with an estimated mass of 0.22 super(+0.03) sub(-0.04) M sub(middot in circle), while NIR photometry of HD 1160 B suggests a brown dwarf with a mass of 33 super(+12) sub(-9) M sub(Jup). The very small mass ratio (0.014) between the A and B components of the system is rare for A star binaries, and would represent a planetary-mass companion were HD 1160 A to be slightly less massive than the Sun.