Singh, who arrived earlier on a two-day visit to the Afghan capital, was given a ceremonial reception by the Afghan Police, the Army and the Air Force in the grand presidential palace ''The Arg''.
"This is a tremendous honour for the Afghan people to welcome His Excellency Dr Manmohan Singh, Prime Minister of India, to second home Afghanistan," said Karzai, welcoming Singh.
The Prime Minister, in turn, described India and Afghanistan as ''partners in progress''.
"It is always a great honour and privilege to visit you. That you have invited me to visit is an honour to entire people of India," Singh said.
"India and Afghanistan are partners in progress and we will stand by you through thick and thin, and we reaffirm our solidarity with the people and government of Afghanistan," he told the Afghan president at the reception at the palace.
The Afghan President also hosted a banquet in honour of the Prime Minister.
The presidential palace, known as the ''The Arg'', houses the president's office, his residence, and the office of the Afghan National Security Adviser.
The presidential complex also houses the home of Afghanistan's last king Zahir Shah, where Prime Minister Singh is staying.
Singh is the first foreign head of government for whom the late king Zahir Shah's home has been thrown open.
Earlier Singh arrived here, accompanied by National Security Adviser Shivshankar Menon, Foreign Secretary Nirupama Rao, Prime Minister's special envoy to Afghanistan Satinder Lamba and other senior officials, on his first visit in six years.
Singh will hold bilateral talks with Karzai during which he will discuss terrorism, regional development and ways to step up India's assistance to the war-torn country.