Many Park supporters waved national flags and shouted "president" as Park's car entered the facility.
Prosecutors can detain her for up to 20 days before formally charging her, meaning she will likely be in jail while her case is heard. A district court normally issues a ruling within six months of an indictment.
The Seoul court's decision is yet another humiliating fall for Park, South Korea's first female president who was elected in 2012 amid overwhelming support from conservatives, who recall her dictator father as a hero who lifted the country from poverty in the 1960-70s despite a record of severe human rights abuses.
Prosecutors accuse Park of colluding with a confidante to extort big businesses, take a bribe from one of the companies and commit other wrongdoing. The allegations led millions of South Koreans to protest in the streets every weekend for months before lawmakers impeached her in December and the Constitutional Court ruled in March to formally remove her from office.
It made Park the country's first democratically elected leader to be forced from office since democracy got there in the late 1980s.
South Korea will hold an election in May to choose Park's successor.
Prosecutors can charge Park without arresting her. But they said they wanted to arrest her because the allegations against her are "grave" and because other suspects involved the scandal, including her confidante Choi Soon-sil and Samsung heir Lee Jae-yong, have already been arrested.