This rule allows no exceptions.
This sounds natural to me, but I am told "The rule admits no exceptions" can be an alternative way to express this. However, some say using admits is not correct or natural.
I would like to ask how you feel it.
Indeed, the correct phrase would be-
This rule allows no exceptions.
The usage of admit in place of allow would sound very odd and is possibly incorrect therefore you should refrain from using it, instead stick with allow.
People who believe that admit can be correct are possibly influenced by the quote:
"No rule is so general, which admits not some exception." - Robert Burton
The context in which Robert Burton said is most probably different to what you think are rules and thus should not be mixed up.
Per my comments to other answers, the overwhelmingly preferred form is admits of...
This rule admits of no exceptions 1270 hits in Google Books
This rule allows of no exceptions 4 hits
This rule admits no exceptions 8 hits
This rule allows no exceptions 10 hits
Obviously there's no difference in meaning - this is purely a matter of established idiomatic usage.
"Allow" is the more common usage.
Alternately to This rule allows no exceptions, you can also say:
This rule allows of no exceptions.
This rule admits of no exceptions.
allow; admit (often followed by of): to permit something to happen: This rule allows of no exceptions; The evidence allows of only one interpretation.
Sources: Cambridge Dictionary and Random House
There are two senses of admit which make perfect sense in this phrase, and I'm honestly not sure which is more relevant.
it could be admit - acknowledge the existence of, so we have
This rule admits (the existence of) no exceptions
which is to say, this descriptive rule claims no exceptions to the rule exist.
or it could be admit - permit entry to, giving
This rule admits no exceptions (entry into the subject under discussion)
which would be a prescriptive rule explicitly disallowing exceptions.
However, they're both entirely valid, if a little old-fashioned