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8 year oldSouth Korean military officials say the two rockets are both powerful, intermediate-range Musudans.
Four other missiles tested in the last two months are reported to have either exploded mid-air or crashed.
North Korea, which is pursuing a nuclear weapons programme, is banned by UN resolutions from any use of ballistic missile technology.
The fate of the second missile tested was not immediately clear.
The Japanese Prime Minister, Shinzo Abe, condemned North Korea's missile testing as intolerable.
Surrounding countries had detected preparations for a launch in the past few days and warned that it was about to happen.
The Musudan has a reported range of 3,000km (1,800 miles) - enough for it to hit South Korea, Japan and the US base of Guam.
North Korea is thought to have about 50 of them, the BBC's Korea correspondent, Steve Evans, reports.
North Korea's opponents had watched with trepidation as the first missile was truck moved into a launching position.
Japan said it would shoot down a missile if its trajectory was over the country.
The failure of the first missile illustrates the technological difficulties North Korea faces as it develops a nuclear arsenal and the means to attack distant targets, our correspondent says.
Repeated testing also illustrates a determination to succeed, he adds.