WASHINGTON — President Barack Obama on Monday said there was “no excuse” for the cascade of computer problems that have marred the rollout of a key element in his health care law, but declared he was confident the administration would be able to fix the issues.

“There’s no sugarcoating it,” Obama said. “Nobody is more frustrated than I am.”

The president said his administration was doing “everything we can possibly do” to get the federally run websites where people are supposed to apply for insurance up and running. That includes bringing in additional technology experts from inside and outside the government to work on the issues.

People have until March 31 to sign up for coverage. The non-partisan Congressional Budget Office had projected that about 7 million people would gain coverage through the exchanges during the first year.

The president on Monday guaranteed that everyone who wants to get insurance through the new health care exchanges will be able to, even if they have to enroll over the phone or fill out a paper application. The White House also appeared to open the door to the possibility that people trying to purchase insurance who were confounded by website problems might be exempted from the law’s penalty for remaining uninsured after March 31.

Obama’s event in the White House Rose Garden had the feeling of a health care pep rally, with guests in the Rose Garden applauding as Obama ticked through what the White House sees as benefits of the law. The president was introduced by a woman who had successfully managed to sign up for health insurance through the marketplaces in her home state of Delaware.

The rollout failures have been deeply embarrassing for the White House. The issues have called into question whether the administration is capable of implementing the complex policy and why senior White House officials — including the president — appear to have been unaware of the scope of the problems when the exchange sites opened on Oct. 1.

Obama, in his most extensive remarks about the health care problems, insisted Monday that the health care law is about more than just a website.

“The essence of the law, the health insurance that’s available to people, is working just fine,” he said during his 25-minute remarks.

The White House says more than 19 million people have visited HealthCare.gov since the site went live on Oct. 1. Officials also say a half million people have applied for insurance on the federal- and state-run websites.

Although administration officials have acknowledged issues with software and some elements of the system’s design, they have repeatedly cited high consumer interest as among the factors overwhelming the website when people try to log on.

“There’s no question that the volume has caused some of the problems but also exposed some of the problems,” White House spokesman Jay Carney said Monday.