The adoption of the US Constitution itself, in 1788, might meet your requirements. After the Declaration of Independence the colonies organized themselves into a nation under the Articles of Confederation. This was good enough to see them through the War of Independence, which ended in 1783. The next war they were involved in was the Franco-American naval war of 1798-1800. There were various tensions and conflicts with Native American Tribes throughout the early history, however, but few of them rose to the level of being officially recognized as "wars". Depending on how you view those issues and conflicts, you might treat the period of 1783-1788 as one of "peace".
Soon after the war of independence, it became apparent that the articles were ineffective and that a stronger centralized government was needed to maintain unity and deal with the issues the states and nation as a whole faced. So the states assembled a constitutional convention, via the procedures of the Articles of Confederation, whose original intent was to amend the articles so as to solve these problems. However, it soon became apparent to the convention that amendments simply weren't going to do, and the entire constitution had to be rewritten and replaced.
So the US Constitution as we would come to know it started to be developed. The notable thing here, which may make it meet your requirements, is that while the proposed constitution only required ratification by 9 of the 13 states to adopt, the Articles of Confederation themselves declared themselves perpetual, inviolable, and that any changes to them required unanimous approval in its Article XIII:
[T]he Articles of this Confederation shall be inviolably observed by every State, and the Union shall be perpetual; nor shall any alteration at any time hereafter be made in any of them; unless such alteration be agreed to in a Congress of the United States, and be afterwards confirmed by the legislatures of every State.
So while the process started out following the terms of their existing constitution, the procedure for adoption of the new one violated it. There was some slight debate and contention on the matter of the legitimacy of the new constitution as a result, but it was not terribly significant. In Federalist #40 Madison even dismissed it out of hand as moot. Some contemperaneous justifications (and still repeated by some modern historians, though it's long since ceased to be a legal issue: by 1790 all states at the time had ratified the new constitution) were that states were already regularly violating the articles, rendering them essentially invalid; that the constitution was a "side deal between states" before ratification that was permitted by the articles; or arguing on a technicality that a complete replacement was not the same thing as an alteration, and so was not covered by the unanimity clause.