Nepal is said to be in shock following the massacre, but no security problems have been reported, although many police have been deployed in the capital Kathmandu.
King Birendra's brother - Prince Gyanedra - has been appointed regent as the crown prince remains gravely ill in hospital after turning the gun on himself.
There has been no official word about what triggered the worst violence in the centuries-old history of Nepal's royal family, with reports suggesting the dispute centred on the assassin's choice of bride.
Nepal is said to be in shock following the massacre, but no security problems have been reported, although many police have been deployed in the capital Kathmandu.
The state funeral is currently taking place in the capital, Kathmandu.
After first reports that the crown prince was also dead, medical sources said he was still alive, gravely ill and on life support.
Succession problems
The BBC's Daniel Lak, in Kathmandu, says this means that whatever the circumstances the constitution and royal tradition make him first in line for the throne.
A long meeting of the privy council - the body that monitors the succession to the throne - named Prince Dipendra as king, but appointed Prince Gyanendra as regent.
Prince Gyanendra is the senior of the late King Birendra's two younger brothers. He was away from Kathmandu at the time of the massacre.
Our correspondent says privy councillors and political leaders were said to be anxious to name Prince Gyanendra as monarch and to try to resolve this crisis.
National tragedy
Our correspondent says there is a mood of deep loss and confusion on the streets of Kathmandu.
"Shocking is an understatement, we have been orphaned by this loss," said a vegetable seller, Janardan Sharma, who like many in Kathmandu rushed to the royal palace early on Saturday to try to find out more news of the tragedy.
The elected Prime Minister and cabinet have been monitoring events closely and are expected to meet shortly to discuss their response to the crisis.
Deputy Prime Minister Ram Chandra Paudel called the massacre "a national tragedy," and confirmed that the crown prince was to blame.
Monarchy has long been considered as the symbol of unity in the multi-ethnic and multi-cultural Himalayan kingdom.
The BBC's Sushil Sharma says Nepalese kings have been credited with success in maintaining the independence of Nepal, which is sandwiched between two of the world's most populous countries and rivals, India and China.
Analysts say Nepal faces a tough challenge in preventing Friday's unprecedented bloodshed from creating instability.
The Crown Prince
Crown Prince Dipendra, educated at exclusive Eton College in the United Kingdom, is reported to have been at odds with his family for some time over his choice of bride.
According to recent press reports, astrologers had advised that the crown prince should not be allowed to marry or have children until he reached the age of 35.
The astrologers warned that the king would die if this advice was ignored.
The king and queen's other two children - Prince Nirajan and Princess Shruti - are among the dead.
The murders are thought to be the worst mass killing of royalty since the Romanovs were put to death by order of Lenin in 1918 during the Russian civil war.
Popular king
King Birendra, 55, ruled Nepal as an absolute monarch after ascending the throne in 1972, without political parties under a system of local panchayats or councils.
But nationwide unrest forced him to legalise political parties in 1990 and accept a parliamentary system.
Some people in Nepal, which is 90% Hindu, believe that the king is the reincarnation of the god Vishnu.