U.S. Immigration and Customs Enforcement officials are on the prowl for illegal aliens (search) living in the United States who have been convicted of serious crimes during their stay.

"In New York, we've got approximately 26,000 fugitives. And those are just individuals who've been ordered deported from the United States by an immigration judge. And of those, approximately 3,000 to 3,500 are criminals," said Ray Simonse of ICE.

Those convicted criminals top the most-wanted list for the New York-based ICE teams, which average at least 20 arrests each week as part of Operation Icecap (search).

One man visited by ICE agents recently immigrated to the United States legally, but a criminal conviction means he's headed for a one-way trip back to Jamaica.

"When they see ICE on the jacket they realize, you know, their day has come," Simonse said.

In most cases, this is the end of the road for illegal immigrants (search) in the United States. ICE agents put illegal aliens on planes back to their native countries in a matter of days or weeks.

ICE investigators say criminal aliens have a high rate of recidivism (search); as many as 45 percent commit another crime. But illegal immigrants often are tough to track down — a frustrating part of the job of the 18 ICE teams that operate nationwide.

Lawmakers from both sides of the political aisle agree that more needs to be done to crack down on illegal immigration (search), particularly in the name of homeland security. Even President Bush, who on Wednesday was meeting with Mexican President Vicente Fox (search) and Canadian Prime Minister Paul Martin (search), said he is concerned about the surge in illegal immigration since the Sept. 11, 2001, attacks.

"There's been a surge because we really haven't done enough to try to curb illegal immigration," Rep. Adam Schiff, D-Calif., told FOX News.

He noted that more border agents need to be posted at the borders, particularly in the south, where there's only one agent per every half mile of border. "That's simply inadequate," he said.

Although 2,000 more agents have been authorized for border patrol, the Bush administration's budget for the upcoming fiscal year only funds 200, Schiff noted. The immigrant-tracking process also can be improved, he said, and more economic incentives can be offered to crack down on employers flagrantly hiring illegal workers.

Rep. Louie Gohmert, R-Texas, said the United States might as well roll out the welcome mat for illegal aliens.

"When we send a clear message, 'we're not going to enforce our laws ... and there may be a chance people can get a free pass for getting here illegally' … [the message sent] is 'come on down,'" Gohmert said.

He said more standards can be established regarding proper forms of identification, and current laws aimed at regulating illegal immigration also can help. But of utmost concern is beefing up border agents, he agreed.

"We need to fund positions along [the] border," Gohmert said. "That, I think, is some of the most effective work we can do. ... We really need to step up to the problem."

Click on the video box above for report on ICE busts by FOX News' Todd Conner.