WASHINGTON—A special enrollment period to obtain health insurance for millions of uninsured people who owe a tax penalty under the Affordable Care Act is off to a slow start.

The health law requires most Americans to have insurance or pay a fine at tax time. The open enrollment period under the health law ended Feb. 15, but the Obama administration said it would allow people who discover they owe a fine to sign up for coverage through April, at the end of the tax season.

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