I believe it is still open. By the Boone-Higman Theorem (W. W. Boone and G. Higman, "An algebraic characterization of the solvability of the word problem", J. Austral. Math. Soc. 18, 41-53 (1974)), a finitely presented group has solvable word problem if and only if it can be embedded in a simple group that can be embedded in a finitely presented group.

It is widely believed that it is possible for the simple group itself to be finitely presented, but (AFAIK) not proved.

So the answer to Agol's comment is that no finitely presented group with unsolvable word problem can be embedded into a finitely presented simple group.