bis spät in die Nacht
Since the whole expression is temporal and uses bis, I would expect it to be in dative, i.e.:
bis spät in der Nacht
However, the accusative form is of common use. Why?
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You have to be careful. I will first focus on the preposition in question, in, then construct a trickier example and finally discuss some general points that were phrased badly elsewhere.
Ich lief bis in den Wald / I ran (all the way) into the forest
signifies an activity transporting me from outside into the forest. By contrast,
ich lief im Wald umher / I ran around in the forest
signifies an activity taking place inside the forest. Now, if you say,
I read well into the night / Ich las bis in die Nacht hinein
I read through the night,
I think you would agree to group this with my first, not with my second example. The second example would correspond to sentences like
In der Nacht las ich / At night I did read
Generally, it is worthwhile to pay close attention to the treacherous difference between:
Ich fiel im Zimmer/In the room I fell
ich fiel ins Zimmer / I fell into the room
until it becomes a second nature!
Let us add one trickier example that may first confuse, but should ultimately help:
Ich las, bis in der Nacht Maurinus kam.
Upon careless glimpse, this might appear to contradict our elaborations. But in reality it means:
Ich las, bis Maurinus in der Nacht kam.
The arrival of Maurinus is contained within the night and that explains the dative!
Often sentences involving bis, will be followed by prepositions, that enable you to deduce the correct case. Often the prepositions will be omitted or admit different cases with vastly different interpretations. Your example is of the last kind.
Now, I shall address some points that are phrased misleadingly in other answers. First, depending on meaning, bis can be followed by any case!
Bis ich genug habe.
Bis des Henkers Schwert brach.
Bis dem Mann ein Licht aufging.
Bis den Schuft der Teufel holt.
Secondly, it is wrong to defer everything to directionality regardless of prepositions, as, it seems to me, the accepted answer does. The phrase
Ich lief bis zum Wald.
is just as directional as
Ich lief bis in den Wald hinein.
It is thus unavoidable to meticulously examine each sentence and in particular prepositions. Only in conjunction with the rest of the sentence, is directionality enlightening!
The reason is that "bis spät in die Nacht" is derived from "bis spät in die Nacht hinein", which literally translates as "until late into the night".
You see that the idea is to describe a direction (in time) rather than a position in time. And to express directions, you use the accusative case in German, just like usual.
The rule is this:
By itself, the expression is "spat in der Nacht"; that is a point (location) in time that uses the dative case.
But with "bis," the construction is "bis spat in die Nacht" (von fruh in der Nacht), which is a direction (in time). That uses the accusative case.
"bis" strongly associates directionality, and German expresses directionality through the accusative. Therefore phrases formed with "bis" use the accusative, whether local
Wir gehen bis ans Ende der Welt.
Wir trinken bis tief in die Nacht.
or completely abstract:
Wir haben nichts zu fürchten bis auf die Furcht.
Each of these prepositions can take dative or accusative, and because of the "bis" it takes accusative in this case.
"bis" can be followed by a dative, but usually only if there is another, stronger reason for having it. For instance, "zu" takes only the dative, so the correct expression is
Wir fahren bis zur nächsten Ausfahrt.
In this case, the rigid syntactic valence of "zu" for a dative overrides the general bias of "bis" for an accusative. ("bis" is unusual in its non-strict case valence. But it's unusual to begin with, since most prepositions take only noun phrases and not other prepositional phrases.)