Terry, 31, was fined £220,000 and banned for four domestic matches for racially insulting Queens Park Rangers defender Anton Ferdinand a year ago this weekend.
Terry will serve the first match of that suspension when he misses the London derby at Tottenham Hotspur in the Premier League on Saturday after deciding not to appeal against the FA's ruling.
Although the FA found him guilty of racially abusing Ferdinand, he was found not guilty of a racially aggravated public order defence by Westminster Magistrates in July.
Buck told radio station Talksport: "John will continue to be captain of the club. We are not going to suspend him for any additional matches that the FA have suspended him for but we have taken firm disciplinary action appropriate to the circumstances."
Asked if Chelsea would try and sell the long-serving Terry in the January transfer window, Buck said they would not.
"We think with John Terry at the club and with John Terry as captain we can move forward from this incident, but we won't forget this incident."
The club's chief executive Ron Gourlay, in the same interview, said that Terry had been given a "very heavy fine" but would not confirm what financial details were involved.
Gourlay added: "The incident 12 months ago has clouded the most successful period in the club's history. We have come down on John very very heavily.
"We have taken into account the punishment handed out by the FA, he has been banned for four games and had a very heavy fine from the FA and the club still felt the need to sanction him more.
"It was a lapse of judgement. It was out of character on the field for John and he did fall below the high standards we expect at the football club.
"We don't believe for one moment that John is a racist and we must not forget he was cleared in a court of law."
Chelsea became European champions for the first time in May, when they beat Bayern Munich in the Champions League final, but Terry missed that match as he was suspended for being sent off in the semi-final against Barcelona.
The two confirmed that the club's billionaire Russian owner Roman Abramovich had played a key part in determining the internal punishment meted out to Terry.
"We consulted with the owner Mr Abramovich and this is not a decision we have taken lightly," Buck added.
"It was not discussed in half-an-hour, it was discussed over a reasonably long period of time and we think we have taken appropriate disciplinary action."
Chelsea manager Roberto Di Matteo continually refused to tell reporters at his pre-match media briefing on Friday whether Terry would remain as captain.
"Robbie could not say that," Buck told Talksport. "It was the Board's decision and not just his decision alone."
- John Terry