Speaking before his detention on Wednesday evening, Mr Adams said he was "innocent of any part" in the murder.
Mrs McConville, a 37-year-old widow and mother-of-10, was abducted and shot by the IRA.
Her body was recovered from a beach in County Louth in 2003.
Police said a 65-year-old man presented himself to officers at Antrim police station and was arrested.
Mr Adams added: "I believe that the killing of Jean McConville and the secret burial of her body was wrong and a grievous injustice to her and her family.
"Well publicised, malicious allegations have been made against me. I reject these.
"While I have never disassociated myself from the IRA and I never will, I am innocent of any part in the abduction, killing or burial of Mrs McConville."
His party colleague Alex Maskey condemned the timing of the arrest, just over three weeks from the European and local government elections.
However, Mrs McConville's son Michael, who was 11 when his mother was murdered, welcomed the arrest.
"We're just happy to see everything moving as it is moving at the minute," Mr McConville said.
"Me and the rest of my brothers and sisters are just glad to see the PSNI doing their job. We didn't think it would ever take place [Mr Adams' arrest], but we are quite glad that it is taking place.
"All we're looking for is justice for our mother. Our mother, on the seventh of next month, would have been 80 years of age.
"Although we didn't spend much time with our mother, we'd have like to have spent a lot of time with her. If the IRA hadn't have killed our mother, God knows, she still might have been alive today."
Mrs McConville, one of Northern Ireland's so-called Disappeared, was kidnapped in front of her children after being wrongly accused of being an informer.
Last month, Ivor Bell, 77, a leader in the Provisional IRA in the 1970s, was charged with aiding and abetting the murder.
There have also been a number of other arrests over the murder recently.
The case against Bell is based on an interview he allegedly gave to researchers at Boston College in the US.
The Boston College tapes are a series of candid, confessional interviews with former loyalist and republican paramilitaries, designed to be an oral history of the Troubles.
The paramilitaries were told the tapes would only be made public after their deaths.
However, after a series of court cases in the United States, some of the content has been handed over to the authorities.
She was held at one or more houses before being shot and buried in secret.
The Disappeared are those who were abducted, murdered and secretly buried by republicans during the Troubles.
The IRA admitted in 1999 that it murdered and buried at secret locations nine of the Disappeared.
The Independent Commission for the Location of Victims' Remains was established in 1999 by a treaty between the British and Irish governments.
It lists 16 people as "disappeared". Despite extensive searches, the remains of seven of them have not been found.
Mr Adams has never been charged with membership of the IRA. He was, however, interned in 1972 under the controversial Special Powers Act, but briefly released in order that he could take part in talks in London between representatives of Sinn Fein and the then Home Secretary Willie Whitelaw.
He was later re-arrested and interned at Long Kesh. Following an aborted escape attempt he received a prison sentence.