MEXICO CITY, Mexico (CNN) -- Reviving a ban on assault weapons and more strictly enforcing existing gun laws could help tamp down drug violence that has run rampant on the U.S.-Mexican border, President Obama said Thursday.

"We can respect and honor the Second Amendment while dealing with assault weapons," Obama says in Mexico.

Speaking alongside Mexican President Felipe Calderón, Obama said he has "not backed off at all" on a campaign pledge to try to restore the ban. It was instituted under President Clinton and allowed to lapse by President George W. Bush.

"I continue to believe that we can respect and honor the Second Amendment right in our Constitution -- the rights of sportsmen and hunters and homeowners that want to keep their families safe -- to lawfully bear arms, while dealing with assault weapons that, as we know here in Mexico, are used to fuel violence," Obama said.

Obama and Calderón spoke after completing a wide-ranging meeting that included talk of the deadly border situation.

Calderón said that the link between Mexican drug violence and the U.S. ban on 19 types of military-style semi-automatic rifles -- which lapsed in 2004 -- is clear.

"From the moment the the prohibition on the sale of assault weapons was lifted a few years ago, we have seen an increase in the power of organized crime in Mexico," Calderón said.

He said that more than 16,000 assault weapons have been seized in the crackdown on drug traffickers, with almost 9 in 10 coming from the United States.

Some observers have said Obama may be slow to reintroduce the ban in Congress, where it would be sure to spark a fight at a time when his administration needs all the political clout it can muster to push its aggressive economic recovery efforts.

Calderón acknowledged the debate's thorny nature.

"We know that it is a politically delicate topic because Americans truly appreciate their Constitutional rights," he said. "As long as we are able to express clearly what our problems in Mexico are, then we might be able to also seek a solution that respects the constitutional rights of Americans, that at the same time will avoid organized crime becoming better armed in our country."

Obama said he has asked Attorney General Eric Holder to study how current gun laws are enforced and whether loopholes in some can be tightened. He said laws already on the books should restrict the flow of weapons into Mexico.

Obama and Calderón said their discussions ranged from working together to combat global climate change, to efforts at comprehensive immigration reform. Obama noted his two votes as a U.S. senator for reforms was backed by then-President Bush but shot down by Bush's fellow Republicans over what they called an "amnesty" provision for illegal immigrants.

"For those immigrants who have put down roots -- they have come here illegally -- I think they need to pay a penalty for having broken the law. They need to come out of the shadows. Then we need to put them through a process where, if they want to stay in the United States, they have an opportunity to earn it," Obama said.

Calderón said the key to reducing illegal immigration is to grow jobs in Mexico, which he pledged to do.

But much of their talk centered on the drug violence. Watch report on President Obama's trip »

Since taking office in 2006, Calderón has worked to root out government corruption and crack down on the drug cartels that hold sway in many of Mexico's border regions. That, combined with ramped-up power struggles and turf warfare, has contributed to a rash of violence that has led to more than 1,000 deaths this year.

Obama commended Calderón on the steps his government has taken.

"But I will not pretend that this is a Mexican responsibility alone," he said. "A demand for these drugs in the United States is what is helping keep these cartels in business. This war is being waged with guns not purchased here but in the United States."

Obama said he'll urge fast-tracking of the three-year, $1.4 billion Merida Initiative, a joint security plan between the United States, Mexico and other Latin American countries in which U.S. equipment, technology and expertise are used toward combating the drug trade.

Speaking to CNN en Español, Obama lauded Calderón as having done "an outstanding and heroic job in dealing with what is a big problem right now along the borders with the drug cartels."

Asked whether the United States is partly to blame for the violence along the border, Homeland Security Secretary Janet Napolitano said, "there certainly is a relationship. You can't deny it."

In Mexico City on Thursday, she said, "What we're working to do is to work to stop the flow of guns and cash into Mexico that are helping fuel these cartels, but also we're working at the border to make sure that the spillover violence doesn't occur in our own cities and communities." Watch what Napolitano says about the U.S.-Mexico drug link »

Napolitano said the United States also must ensure that it is enforcing immigration laws on employers who "consistently go into that illegal labor market in order to exploit it."

E-verify, an electronic employment eligibility verification system, must be an integral part of immigration enforcement, she said.

Obama is to travel later in the week to the summit for meetings with Latin American leaders. While on the trip to Latin America, Obama said he seeks to engage in talks with the region's leaders as equals.

"Times have changed," Obama said Wednesday.

He refused to criticize the leaders of Bolivia, Ecuador and Venezuela, who have taken measures to change their constitutions to extend their holds on power. Watch as President Obama arrives in Mexico »

"I think it's important for the United States not to tell other countries how to structure their democratic practices and what should be contained in their constitutions," he said. "It's up to the people of those countries to make a decision about how they want to structure their affairs."

He said he believes that the United States has a leadership role to play in the region, but he qualified that role this way: "We also recognize that other countries have important contributions and insights."

He added, "We want to listen and learn as well as talk, and that approach, I think, of mutual respect and finding common interests, is one that ultimately will serve everybody."



CNN's Ed Hornick and Kristi Keck contributed to this report.

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