The darkest time of night is midnight. This is the point exactly halfway between dusk and dawn, or sunset and sunrise, when the sun is at exactly 180 degrees to the observer's position.
Of course, other factors must be taken into consideration as well, including moon phase and position. A full moon at extreme perigee, which is the closest orbital distance to the Earth, will naturally be brighter than a new moon, or a moon at extreme apogee, which is the furthest orbital distance from the Earth.
The idea that night is darkest just before dawn is, false. It comes from a well-known quote by the theologian Thomas Fuller (1608-1661): 'It is always darkest just before the day dawneth.' This quote is also often wrongly attributed to the Bible.