Ok, then when do I stop referring to the time after midnight as night and start calling it morning?
Well it's "три часа ночи" and "пять часов утра". When it comes to 4 a.m. opinions differ, but it's usually considered to be morning.
That varies with context. "We stayed up really late last night. We didn't go to bed until one in the morning."
"one in the morning" or "one a.m." happens during the night. "Night" is a generality which means "when it's dark out, after the sun goes down and before sunrise", and the morning starting during the night has to do with how hours are measured - a.m./p.m. Even "3 in the morning" is still "at night" - but once you get to pre-dawn hours, like 4 a.m., that can qualify as morning - or night, depending.
1:00am is without question 'night', no matter which language it is. There are arguments for considering it "morning" as well, but it's easy enough to see: look at the sky at 1:00am; is the sun out? If the answer is "no", then it is nighttime. As a native English-speaker, I have only ever referred to 1:00am as "one o'clock at night".
What about the "one" in the "one o'clock"? Shall we assume that this is implied by час?
Seems like there is very little else to base it on. I suppose that here час means "hour", and since it's in nominative case, it can only mean "one", since all the other numbers in Russian require genitive (although some are declinable).
Ночи = night
It makes absolutely no sense to not accept night as a translation.
A chart of prepositions have says that object of в take prepositional case when used in expressions of time. If that were applicable here then the sentence should be в (одной} чаше ночи.
В takes accusative when motion is involved, so that would work here - but I can't see any logical to that.
Is this some sort of idiomatic exception to the usual rules about the case of objects of в?
But prepositional for, e.g., months, days, years?
Seems like the chart of prepositions I have might be wrong about "expressions of time" - that that should be under accusative rather than prepositional.
In Genitive if I'm not mistaken, in this case it means "of the night". "в час ночи" -> "at one o'clock in (of) the morning".