SEOUL—In a Stockholm cafe one winter’s day in 1999, North Korea’s ambassador to Sweden made a proposal to his Israeli counterpart, according to a former Pyongyang diplomat: Give North Korea $1 billion in cash and we will scrap our agreements to sell our missile technology to Iran and other enemies of Israel.

The Israelis refused, and days later offered food aid instead, according to the account. The talks ended without an agreement. Since then, North Korea has remained a steady supplier of conventional and ballistic weapons...