President Trump said on Friday that the summit meeting with North Korea on June 12 in Singapore had been rescheduled after a week of meetings and scrambling between American and Korean officials.

The president made the announcement after a North Korean envoy, Kim Yong-chol, personally gave him a letter from Kim Jong-un, the North Korean leader. Mr. Trump said he had not yet opened the letter, but felt the meeting — “really a get-to-know-you kind of a situation,” he said afterward — went well.

Secretary of State Mike Pompeo met with Kim Yong-chol, one of the highest-ranking North Korean officials, on Thursday in New York and had expressed optimism this week about the potential talks.

But that same day, Kim Jong-un met with a top Russian official, Sergey V. Lavrov, in North Korea. Mr. Lavrov’s welcome in Pyongyang, the capital — and the invitation he brought from President Vladimir V. Putin for Mr. Kim to visit Moscow — was a reminder that competing powers could still upend efforts to hold the summit meeting.