When Facebook cofounder Eduardo Saverin famously renounced his U.S. citizenship, the move helped him potentially avoid a fortune in capital gains taxes. And while such renouncements are rare, a small but growing number of Americans also are trading in their citizenship for -- at least in part -- financial gain.

Internal Revenue Service records show that 1,001 U.S. citizens gave up their citizenship in the first quarter of the year. That's more than the total renouncements in 2012 and about one-third of 2013's recording-breaking 2,999, according to the Tax Foundation, a Washington think tank that promotes lower taxes.

'They just get so frustrated with the complexity of the rules that they just throw up their hands and say, is it really worth staying a U.S. citizen?'

Most of the individuals are wealthy, although they're not billionaires like Saverin, tax lawyers say. And in most cases they're motivated by a desire to lessen their tax load and are frustrated over a U.S. tax code they say unfairly targets Americans living abroad.

The 2010 Foreign Account Tax Compliance Act has been blamed partly for the recent increase in Americans renouncing their citizenship. The law is designed to go after U.S. taxpayers purposely hiding assets in offshore accounts and shell corporations. But tax lawyers say that normally law-abiding Americans with foreign assets and bank accounts often find themselves targeted by the IRS.

"This is a massive effort on the part of the Obama administration, using all levers -- the Justice Department, the IRS, etc. -- to go after Americans living abroad with some money," said William McBride, chief economist at the Tax Foundation.

Andrew Mitchel, an international tax lawyer in Connecticut, said many of his overseas-based clients who hold U.S. passports are unaware that, unlike most countries, the U.S. requires its citizens who live permanently abroad to pay U.S. taxes.

Those caught by the IRS face penalties of at least $100,000.

"Some of these people, they're willing to pay the fee but then they just get so frustrated with the complexity of the rules that they just throw up their hands and say, is it really worth staying a U.S. citizen?" Mitchel said.

But other countries are starting to adopt laws similar to FACTA, and the U.S. has significantly lower tax rates than much of the industrialized world, so some tax experts don't expect a flood of U.S. citizens to give up their rights as Americans.

"Where are you going to move that has a lower tax rate?," said Edward Kleinbard, a University of Southern California professor and tax law expert. "Yes you can move to Singapore if you want to live in Singapore for the rest of your life, but not if you want to live in Europe, not if you want to live in Canada."

Among the most high-profile Americans to give up his U.S. citizenship was the late businessman and philanthropist John Templeton, who did so in 1964 and moved to the Bahamas. The move allowed him to save millions of tax dollars selling international funds.

Songwriter Denise Eisenberg Rich turned in her passport in 2011 and now lives in Austria. Her ex-husband Marc Rich, a wealthy financier and businessman, was indicted on tax evasion charges and later controversially pardoned by President Bill Clinton.

But Kleinbard said most Americans who renounce their citizenship have "very attenuated connections with the United States." They typically are dual citizens already living abroad, including children of U.S. citizens who have grown up outside the United States, or spouses of U.S. citizens who long ago married and moved abroad.

Saverin abandoned his citizenship before an initial public offering of Facebook stock that was estimated to make him almost $4 billion. And his move to Singapore — which doesn't tax dividends, capital gains, estate duty, and foreign sourced income – was poised to significantly cut his tax bill.

The Harvard-educated entrepreneur fits the description of many others who abandon the U.S., including living much of his life abroad and holding dual citizenship with his native Brazil.

"It's very few cases of people who are fundamentally rooted in the United States uprooting themselves," Kleinbard said.

• An earlier version of this article incorrectly identified Denise Eisenberg Rich's current country of residence.