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At the Convention of States Project's website, and under their Frequently Asked Questions the second question presented and then answered is:
What is a Convention of States?
"A convention of states is a convention called by the state legislatures for the purpose of proposing amendments to the Constitution. They are given power to do this under Article V of the Constitution. It is not a constitutional convention. It cannot throw out the Constitution because its authority is derived from the Constitution."
The Convention of States Project is absolutely correct that when a convention is called, it is for the specific purpose of “proposing Amendments” [that’s plural], see Article V! But their opinion which then follows is not based upon historical facts and precedent. The fact is, the Convention of 1787, which is the precedent we must rely upon, was called for “the sole and express purpose of revising the Articles of Confederation”. The Delegates to the Convention were not authorized to draw up an entirely new Constitution and new government which they took it upon themselves to do. And after doing so, the Convention also decided to ignore the Articles of Confederation's requirement that a unanimous consent be obtained by the States to alter the Articles. The Convention changed that requirement to an approval by a mere nine states to put into effect the new government and new Constitution they created.
And so, it is disingenuous for the Convention of States Project to make the claim a convention, if called, “cannot throw out the Constitution”. It certainly does have the constitutionally authorized power to propose countless “amendments” which in effect could eviscerate our entire Constitution and created a new system of government if adopted. And judging from historical precedent, the Convention may also propose a method of adoption different than what our existing Constitution currently requires.
It should also be noted that a Justice of the Supreme Court has already confirmed the autonomy of a convention should one be called. Chief Justice Warren Burger wrote to Phyllis Schlafly in 1988, regarding another convention: “ I have also repeatedly given my opinion that there is no effective way to limit or muzzle the actions of a Constitutional Convention. The Convention could make its own rules and set its own agenda. Congress might try to limit the Convention to one amendment or to one issue, but there is no way to assure that the Convention would obey. After a Convention is convened, it will be too late to stop the Convention if we don’t like the agenda. The meeting in 1787 ignored the limit placed by the Confederation Congress ‘for the sole and express purpose.’
JWK
"What about a runaway convention? Yes, it is true that once you assemble a convention that states have called, they can do anything they want." ___ Virginia’s Attorney General Ken Cuccinelli