Let me point out that that was an act of the Texas government, not the U.S. government. I would also point out that Article IV, Section 3, Clause 1 states, "...no new State shall be formed or erected within the Jurisdiction of any other State; nor any State be formed by the Junction of two or more States, or Parts of States, without the Consent of the Legislatures of the States concerned as well as of the Congress.
Texas could have split before admission. But once in they have to get the OK from Congress to do it.
And where is that secession clause again?
Those were conditions set by the Republic of Texas, so obviously the US Congress accepted them. I don't know where the secession clause is, but I will try to find it. Maybe in another document.