Loretto school in Musselburgh came under fire from the Office of the Scottish Charity Regulator (OSCR) last year for not providing enough financial support for pupils from poorer families, and was at risk of losing its status.
The school was deemed not to have appropriate systems in place to widen access to pupils from less privileged backgrounds, and given 18 months to pass the test set by the regulator.
Eight other schools were also assessed and retained their charitable status, which allows for tax and rates relief.
Loretto, whose former pupils include Better Together leader Alistair Darling and BBC journalist Andrew Marr, has announced its intention to offer bursaries of 105% of the school fees, which are up to £19,440 per year for day pupils and £28,590 for boarders, to students enrolling from September this year.
Previously bursaries covering only a percentage of the fees were offered, and it is hoped the new scholarship will help with the cost of school uniforms and extra-curricular activities.
Elaine Logan, acting headteacher of Loretto School, said: "The announcement of this bursary underlines our commitment to children being able to realise and fulfil their potential at Loretto, whatever their family's financial circumstances.
"Loretto is the only private school in Scotland to offer a 105% bursary, and it demonstrates that we are acutely aware of the financial pressures families can be under during a child's education. We are extremely fortunate to be able to offer these exceptional bursaries to children who would benefit from such an education, but whose family may not otherwise be able to afford it."