Does the 14th Amendment's due process clause prohibit the suspension of Habeas Corpus?

The constitution provides that the Great Writ can be suspended in times of rebellion or insurrection, when it is necessary for public safety.

However, the 14th Amendment came along after that part was written, and says that no person shall be deprived of life, liberty, or property without due process of law.

So when it comes right down to it, would they still have the authority to suspend the Writ?

The 5th amendment already had the wording, as applied to the federal government, so no. Think of the 14th as making the 5th (and subsequently other parts of the BoR) applicable to the states and not just to the feds. Look at the text:

Emphasis added

You only need to look at whether one constitutional provision comes later in time than others when they actually conflict. As John notes, the Fifth Amendment was adopted contemporaneously with Article I, so clearly the Founders did not believe the Fifth Amendment prohibited the suspension of habeas corpus. Hence, no conflict, even if the Fourteenth Amendment applied to the federal government.

The Bill of Rights were ratified after the Constitution was adopted. So is it not possible that the States had changed their mind on the suspension of habeas corpus within those (I believe) two years?

You’re skipping a whole lot of analysis that comes before that.

Basically, if there are conflicting provisions in law, the first question is whether one can read them both to have equal effect – as in, do these laws make sense if one does not override the other? The answer in this instance is clearly yes: one can read the 14th Amendment in a reasonable way so that it does not eliminate the suspension of habeas. The analysis should end there.

If there were no logical way to read both provisions to give them full effect, then the more specific provision is preferred over the more general – in this case, suspension of habeas would surely still survive.

If there were still no clear answer after that analysis, then one can turn to the “later in time” principle.

I didn’t skip anything. You have to find a conflict before you need to determine how to resolve the conflict.