It is the first time a member of the US cabinet has visited the secretive communist state and the first time President Kim has met an American.
It is the first time a member of the US cabinet has visited the secretive communist state and the first time President Kim has met an American.
There are no details yet of the discussions, which have already lasted more than an hour.
If Mrs Albright's visit is successful, President Bill Clinton is expected to go to North Korea soon.
A BBC correspondent travelling with Mrs Albright says her presence in Pyongyang at the same time as the visiting Chinese defence minister, Chi Haotian, is symbolic, since their countries fought on opposite sides in the Korean War 50 years ago.
The US delegation does not expect to sign any agreements during the visit, but Mrs Albright will recommend for or against a visit by President Clinton before he leaves office in January.
The US still keeps 37,000 troops in South Korea, which has still to sign a formal peace treaty with the North.
Food and dance
Mrs Albright began her visit by meeting President Kim's deputy, Vice Marshall Jo Myong-rok.
According to a US official, Marshall Jo was anxious that Mrs Albright's visit should pave the way for one by President Clinton.
Mrs Albright stressed the importance of North Korea
addressing US concerns before such a trip could be possible, the official said.
Mrs Albright then visited a kindergarten that also serves as a distribution point for World Food Programme assistance.
She danced with brightly-dressed infants, who put on a performance for her.
She joked that she was the first US secretary of state to dance with the children of North Korea.
But she also sent an indirect message to the North Korean leadership, which has been accused of redirecting food aid to its armed forces.
"International donors should be assured that the supplies they send are used for the purposes intended," Mrs Albright said.
Preparing the way
More than 40 US officials have been in Pyongyang since last week preparing an agenda and security for Mrs Albright's arrival.
"We want to see our concerns about North Korea's nuclear and missile programmes addressed and to support the progress the two Koreas are making toward greater contact and warmer relations," she said in a speech on Thursday.
She is to brief the South Korean and Japanese foreign ministers about her trip in Seoul on Wednesday.
North Korea, plunged into a severe economic crisis and food shortages, has taken dramatic steps toward ending five decades of self-imposed isolation since an historic summit in June between Kim Jong-il and President Kim Dae-jung of South Korea.
In the last few months there has also been a remarkable effort by North Korea to reach out diplomatically to the rest of the world.
Pyongyang has given unprecedented access to international journalists to cover Mrs Albright's visit.