The man, aged 29, was detained in Leitrim at 5pm in connection with Mr Black’s murder in Co Armagh on Thursday.
Secretary Clinton said she applauded the swift efforts being made to bring the perpetrators to justice.
“There is no justification for this outrageous and cowardly act. I offer my sincere condolences to the family, friends and colleagues of Officer Black, who had a long and distinguished record of service. The United States remains resolute in support of the people of Northern Ireland, who have condemned violence and embraced the path to peace and reconciliation,” she said.
Gardaí said: “The man is detained in Carrick-On-Shannon Garda Station under Section 30 Offences Against the State Act 1939.”
Earlier in the day, Taoiseach Enda Kenny had pledged full co-operation with the PSNI into the investigation of the “cold blooded murder” of Mr Black as a prominent dissident republican was also arrested by investigating officers.
Colin Duffy, 44 and a second man, 31, were detained by officers in Lurgan, Co Armagh, just miles from where Mr Black, 52, was assassinated in a drive-by shooting on the M1 motorway.
Duffy was acquitted by a judge in Belfast earlier this year of the murders of two soldiers shot dead by dissident republicans outside Massereene military barracks in Antrim in March 2009.
Mr Black, from Cookstown, Co Tyrone, was shot dead as he drove to work at Maghaberry Prison, near Lisburn, Co Antrim, on Thursday morning.
Mr Kenny, at a meeting of the North-South Ministerial Council in Armagh, said the State would provide full support to the murder probe. “Any assistance the PSNI need from the gardaí and the Government will of course be immediate forthcoming — absolutely.”
Asked if he was concerned that the car believed to have been used in the assassination had Dublin registration plates, he replied: “I understand that it was a ’94 registration car that possibly was stolen in Dublin. We’ll wait and see what the full details of the PSNI-Garda report will be.”
The car, a Toyota, 94 D 50997, was found burnt-out in the Inglewood area of Lurgan.
Mr Kenny said people were well aware of the “rise of dissident threats”, and that cross-border co-operation was “fundamental” in combating the problem.