WASHINGTON—North Korea’s only World Health Organization-certified manufacturer of medicines faces closure within weeks if the United Nations doesn’t give the company an exemption from new sanctions, the company’s top officials said.

Pyongsu Pharma was already struggling to produce basic drugs as international sanctions escalated, but a new U.N. resolution approved in September banning foreign joint ventures with North Korea will force the company to close if it doesn’t secure a special allowance. Pyongsu officers said the...