There is currently no US Embassy in North Korea, but that does not mean that there are no diplomatic relations between the two countries, which there are. Of course how well the US and North Korea have gotten on has changed drastically over time.
As to why there is no American Embassy in Pyongyang, essentially the US supported and recognized the Republic of Korea (south Korea) rather than the Democratic Peoples Republic of Korea (north Korea).
You can read bout what embassies are in North Korea here.
Does the United States recognize North Korea?
For all intents and purposes the United States does recognize that existence of the DPRK/ North Korea, although it does not have diplomatic relations with them, or visa versa.
Contact therefore has to be done through official and semi-official channels. Currently the only county in the world that fully does not recognize North Korea is in fact south Korea.
Who looks after Americans in North Korea?
Due to a lack of US embassy in North Korea consular services have traditionally been offered by Sweden who have helped with people arrested such as Kenneth Bae.
As we have seen ourselves this means Sweden looking after the cases and then communicating with the United States on their behalf.
Are there any other contacts between the US and North Korea?
Contacts and communication between the US and North Korea is also done at United Nations to negate the problems of there being no US embassy in Pyongyang.
These are done in New York via the DPRK’s Permanent Representative to UN namely Kim Song.
Will there be a US Embassy in North Korea?
During the famed Trump-Kim summit there were big talks about opening perhaps not an embassy, but a “special office” much like they had done in Cuba and to an extent Taiwan, a pseudo-recognition of you will.
Sadly this was never to come to pass and of course Americans cannot actually visit North Korea now.
The rest of y’all can though, by joining one of our North Korea Tours.