"Hard though it may be we must remain patient and allow the law to take its course."
"After nearly five years of searching it looks as if we have been able to make progress," said Caroline's father John Dickinson.
But Mr Dickinson said the family was remaining cautious and still prepared for a long haul in the fight for justice for Caroline.
He was speaking just minutes after French investigators announced that DNA samples taken from a Spanish suspect had "very important similarities" to samples taken from the scene of Caroline's murder.
Caroline, 13, from Launceston in Cornwall, was attacked on a school trip in France in 1996 while she was staying in a youth hostel dormitory.
Mr Dickinson said: "This was such a horrendous crime, we could never give up hope."
Praise for US officer
However, he said the family had to remain "extremely cautious" about what it could say ahead of further legal proceedings.
"It may be several months before we can say that justice for Caroline has been done.
"Hard though it may be we must remain patient and allow the law to take its course."
He thanked US immigration service official Tommy Ontko, who made the connection to Caroline after Mr Montez was arrested in the US on an unrelated charge.
The US authorities only made the connection to the Dickinson case because Mr Ontko, while on holiday in Europe, read a British newspaper article in which Mr Montez was named, and checked federal files.
"It would appear that international coverage of the investigation...may have led to this fantastic breakthrough being made," said Mr Dickinson.
The family has appealed to international governments to improve their co-operation over serious crime investigations.
In particular it has urged authorities to share DNA information and to set an international standard for DNA testing.
Asked if the family would go to France to see any murder trial first hand, Mr Dickinson said: "We will probably, without doubt be there."
He added the last five years had been "incredibly stressful" and thanked those who had stood by them through "harrowing and very difficult times".