House Speaker Nancy Pelosi said it was with “profound sorrow” that she opened a private service in her honour on Friday.
Democratic presidential nominee Joe Biden and his wife, Jill, sat quietly with other elected officials, friends and family.
His vice-presidential running mate, Senator Kamala Harris, also attended.
Mourners gathered under coronavirus restrictions for the service for Ms Ginsburg, who died last week at age 87, as her casket made the short procession from the court’s steps where it had been on public view for several days to the East Front of the Capitol.
A military honour guard carried it inside the Capitol’s Statuary Hall.
Ms Pelosi and Senate Democratic leader Chuck Schumer welcomed Ms Ginsburg’s casket with the Capitol in turmoil.
President Donald is prepared to announce a conservative nominee to replace the justice on Saturday, weeks before the election. A Senate confirmation vote is expected in late October.
Speaking ahead of the event, Ms Pelosi told CBS Americans need to know what is at stake for the “rush” to confirm Ms Ginsburg’s replacement.
Friday’s ceremony began as a celebration and honouring of Ms Ginsburg’s life and work, with musical selections from one of her favourite opera singers, mezzo-soprano Denyce Graves.
She is the first Jewish American to lie in state at the Capitol.
Ms Ginsburg was remembered as a bright Columbia graduate who was passed over for jobs at a time when few women entered law, only to go on to reshape the nation’s laws protecting women’s rights and equality.
To many she was simply RBG or The Notorious RBG, as legions of fans called the Brooklyn-born justice.
Rabbi Lauren Holtzblatt, of the Adas Israel Congregation in Washington, said “brick by brick, case by case” she changed the course of American law.
“Today, she makes history again,” the rabbi said.
Few Republicans attended the service, which was filled with women and Democrats.
Senators Elizabeth Warren, of Massachusetts, and Amy Klobuchar, of Minnesota, both former presidential contenders, were among those attending.
Senate Majority Leader Mitch McConnell, who is leading the push for Mr Trump’s nominee to replace Ms Ginsburg, was invited but noticeably missing.
GOP whip Steve Scalise, of Louisiana, was there.
Services were brief, with the rabbi’s reflections and prayer coming before guests lined up to pass by the casket and pay their respects.
Towards the end of the line of mourners, one dropped to the ground and did three quick push-ups.
It was Bryant Johnson, the justice’s beloved trainer of her popular workouts.
Members of the House and Senate who were not invited to the ceremony because of space limitations imposed by the coronavirus pandemic will be able to pay their respects before a motorcade carrying Ms Ginsburg’s casket departs the Capitol early in the afternoon.
The honour of lying in state has been accorded fewer than three dozen times, mostly to presidents, vice-presidents and members of Congress.
John Lewis, the civil rights icon, was the most recent person to lie in state after his death in July.
Henry Clay, the Kentucky politician who served as Speaker of the House and also was a senator, was the first in 1852.
Rosa Parks – a private citizen, not a government official – is the only woman who has lain in honour at the Capitol.
Ms Ginsburg has lain in repose for two days at the Supreme Court, where thousands of people paid their respects, including Mr Trump and first lady Melania Trump on Thursday.
Spectators booed and chanted “vote him out” as the president, who wore a mask, stood silently near her casket at the top of the court’s front steps.
Mr Trump plans to announce his nomination on Saturday of a woman to take Ms Ginsburg’s place on the high court, where she served for 27 years and was the leader of the liberal justices.
Ms Ginsburg, the second woman to serve on the Supreme Court, will be buried next week in Arlington National Cemetery beside her husband, Martin, who died in 2010.
