The move, by St Aloysius College in Glasgow, means current S2 pupils will skip the first year of new National 4 and National 5 exams when most of the rest of the country will sit them.
The secondary – which regularly sits near the top of exam league tables – also said it had no intention of adopting the "broad general education" demanded under the Curriculum for Excellence (CfE).
The decision will re-ignite concerns over the erratic nature of the implementation of CfE, and its associated exams, being developed by the Scottish Qualifications Authority.
Earlier this month, there was a national outcry after East Renfrewshire Council became the first local authority to delay the new exams for a year in all its schools, with many arguing it gave pupils there an unfair advantage.
In addition, East Renfrew-shire has rejected the idea that exam choices for pupils should be made at the end of S3 to allow them more freedom to learn without the pressure of exams.
Instead, they will continue to encourage pupils to make exam choices at the end of S2, spending two years working towards the new National 4 and National 5, which replace Standard Grades and Intermediates.
In the wake of the move, the Scottish Government provoked further anger after claiming East Renfrewshire was a "special case".
John Stoer, headmaster of St Aloysius, which was founded in 1859, said the delay would allow the school "considerable time to plan and prepare for the new courses".
He added: "In terms of overall curriculum design, we have no intention of moving, as proposed in CfE, to broad general education until the end of S3.
"We will continue with our existing structure of a broad general education until the end of S2 at which point pupils will make their option choices.
"I am convinced this will put our young people in a stronger position when applying to university, in this country and abroad, and will allow them to acquire a broad range of qualifications before they chose their Highers."
Larry Flanagan, general secretary elect of the Educational Institute of Scotland teaching union, said there should be flexibility for all schools in both the state and private sector.
"Every school should have the right to do this and, if the private sector can do it and the public sector can't, then it highlights the inequity of the system," he said.
"There should be a level playing field for all schools and they should be allowed to make a decision based on where they are at in terms of these new exams."
John Edward, director of the Scottish Council of Independent Schools (SCIS), said it was a strength of the private system that schools could choose what was best for them.
"Schools recognise the dual running option is available and those that want to stay with the old system will do it, but there isn't the same view across the whole sector," he said.
"Parents and pupils expected that flexibility of approach and for the school to make a decision based on what is in the best interests of pupils."
The Scottish Government believes there is no need to offer a wider delay because no other councils wish to postpone the new exams.
On the issue of when schools choose their subjects, Education Scotland has been asked to monitor the situation closely.