Relinquishment of US citizenship under US law is governed by Section 349 of the Immigration and Nationality Act, codified at 8 USC 1481. This law lists a series of relinquishing acts that must be performed with the intention of relinquishing US nationality. One of these relinquishing acts is to make a "formal renunciation of nationality before a diplomatic or consular officer of the United States in a foreign state..."
Note that losing US citizenship under immigration and nationality law may not be sufficient to lose US citizenship for tax purposes. See The United States of America – One country two citizenships – Introducing the "Tax Citizen" and Are you a US citizen? The US has two kinds of citizenship: Citizenship for tax purposes and citizenship for nationality purposes
Blog posts that will be of interest to those contemplating renunciation of US citizenship (note that these posts focus on the nationality aspects of renunciation, not the tax aspects):
- Renouncing US citizenship is just one way of relinquishing US citizenship
- How to Renounce/Relinquish Isaac Brock Society page with links to many useful discussions about the renunciation and relinquishment process
- Those renouncing may find the Consulate Report Directory useful in determining the waiting time and procedures at specific consulates.
- Considering renouncing citizenship? #citizide: Non-citizens can be be inadmissible to the United States by statute
- Considering renouncing US citizenship? #citizide – There are times when US citizenship can save you from foreign taxes!
- Considering renouncing US citizenship? @Expatriationlaw interviews Immigration Attorney @JoeGrasmick
- Global Entry, NEXUS and the effect of renouncing US citizenship
- Synopsis: Solving U.S. Citizenship Problem 
- How to get rid of your U.S. citizenship 
Blog posts that will be of interest to those who wish to document a prior relinquishment:
- Read These if Relinquishing Act Performed Prior to June 4, 2004 A collection of links to posts on the Isaac Brock Society site about prior relinquishments.
- Renunciation is one form of relinquishment – It’s not the form of relinquishment, but the time of relinquishment Explains the difference between citizenship under nationality law and under tax law and why the date of relinquishment matters
- Q. Is a CLN necessary to relinquish US citizenship for tax purposes? A. It depends on the date of relinquishment Explains the role of the Certificate of Loss of Nationality (CLN) in documenting loss of nationality and the date of such loss
- The $2350 "relinquishment fee" does NOT mean that people should simply renounce citizenship - Discussion of the difference between renunciation and other forms of relinquishment
- Americans Abroad And Voting Part 2: Born in the USA? Those who relinquished US citizenship under INA 349(a) are NOT eligible to vote in the November 3, 2020 US election Choosing to vote in 2020 could indicate a lack of intent to relinquish US citizenship in the past
Other blog posts of interest: