WASHINGTON (Reuters) - The U.S. special representative for North Korea, Stephen Biegun, said on Monday that "diplomacy is still very much alive" with North Korea, despite the collapse of a summit between U.S. President Donald Trump and North Korean leader Kim Jong Un last month.

But Beigun, speaking at a nuclear conference in Washington, said gaps remain between the two sides and North Korea must show it is fully committed to elimination of its nuclear weapons.

"It's certainly our expectation that we will be able to continue our close engagement," he said, while offering no specifics on when new talks might be held.

Trump and Kim's second summit in Vietnam last month broke down over differences on U.S. demands for Pyongyang's denuclearization and North Korea's demand for sanctions relief.

Biegun said the United States did not know whether Kim would decide to conduct a new missile launch but reiterated Trump's assertion that he would be disappointed if North Korea were to resume testing.

He said the Trump administration takes "very seriously" reports about new activity at North Korea's Sohae rocket site and was monitoring the situation closely. But he cautioned against drawing any snap conclusions.

U.S. think tanks and Seoul's spy agency said last week that North Korea was rebuilding a rocket launch site. Non-proliferation experts have said satellite images indicate North Korea could be preparing to launch a missile or a space rocket in spite of a freeze in testing that has been in place since 2017.

(Reporting by David Brunnstrom and Matt Spetalnick; editing by Grant McCool)