Kim, who died Tuesday of pneumonia at age 85, was a champion of democracy and inter-Korean reconciliation.
"Your footprints will forever remain as part of our proud history," South Korea's Yonhap news service quoted Prime Minister Han Seung-soo as saying in his eulogy.
North Korea sent six special envoys, including the secretary of the powerful Workers' Party, Kim Ki-nam, Yonhap said.
Former U.S. Secretary of State Madeleine Albright, who led a delegation of 10 to the funeral, remembered Kim as a good friend and inspiring leader.
"President Kim Dae-jung was an amazing leader who set a good example to people in Korea and also to the global community, Albright told Yonhap. "He was a good friend and it's a great loss."
Kim, who took office in 1997, was the first South Korean president to hold a summit with North Korean leader Kim Jong-il in 2000. His efforts at inter-Korean reconciliation won him a Nobel Peace Prize that year.