Is there a term for the period between midnight and sunrise?
The term you are probably looking for is the small hours.
Collins defines this term as "the early hours of the morning, after midnight and before dawn."
In military (US) slang that period is referred to as "oh-dark hundred" or sometimes "zero-dark hundred". On the 24-hour clock the hours before 10 am start with a 0; so 1:00 am is 0100 and said as oh-one-hundred and so forth. Thus oh-dark hundred is anytime after midnight while it is still dark:
"They woke us up at oh-dark-hundred and ran us thru the obstacle course." means:
they woke us up in the wee hours of the morning before daylight.
Short answer: No. There is no common English term for the dark hours of the day that's appropriate at 12:30 AM and not appropriate at 11:45 PM.
I'm not in the military, so I'm not an expert on "oh dark hundred", but I have heard the term (and the term "oh dark thirty" which means exactly the same thing) on many occasions and would consider it inappropriate for 12:30 AM. It's closer to "unpleasantly early in the morning" than to a technical term meaning "the dark hours after midnight".
You can consider postmidnight. It is used as an adjective, so you would say postmidnight hours or postmidnight period.
After midnight, but generally before dawn [Wiktionary]
Examples:
On tests using flight simulators, pilots make more and larger errors when flying during the postmidnight hours.
Flying Magazine Feb 1984
I started out working the graveyard shift—the postmidnight hours that the guys with more pull and experience are eager to avoid.
Blood Secrets: Chronicles of a Crime Scene Reconstructionist By Rod Englert, Kathy Passero
German has a word for this also: Nachmitternacht. It is defined as "the time from midnight to morning" in A Dictionary of the German and English Languages (by George J. Adler). In English, I see the compound form aftermidnight also but it is not that common.
That bit is called 'night'.
OED " the time between evening and morning."
http://www.oed.com/view/Entry/126965?rskey=JeRvu9&result=1#eid
Objectively, evening lasts until midnight, and morning begins at dawn. Subjectively, evening is between sunset and going to bed, night is while you sleep, and morning is when you wake up.