"with only one exception" is a correct and usable phrase in written English.
You can use it when you want to emphasize a particular exception to a rule or general understanding. For example, "Everyone in the class has completed the assignment, with only one exception."
The number of people donating has risen year-on-year, with only one exception, over the last decade from 709 in 2003 to almost double that last year, NHS blood and transplant (NHSBT) figures show.
From column 3, it is clear that German estimates, based on matching procedures (with only one exception), account for this distribution.
What makes this article unique compared to other articles is that the cards listed below (with only one exception) feature 0% introductory rates on balance transfers AND new purchases.
With only one exception, there is now no place on earth where Jews are endangered for being Jewish.
The majority of studies that incorporated direct dual-task training demonstrated improvement on some aspect of dual-task postural control, with only one exception.
Conclusions that are most consistent with the binding data indicated a single binding site on most proteins, irrespective of the probe or the protein, with only one exception noted so far.
With only one exception, the cameras submitted for this roundup cost around $250 or less.
(It probably should be noted that in the premiere episode, with only one exception, the people approached by the comedians are white).
Each paper carried seven letters and, with only one exception, the writers showed no interest in Miller's overseeing of press regulation.
But since 1831, with only one exception — the Whigs in 1836 — every major party has nominated its Presidential candidate at a Convention.
