The most common and most likely answer to this crossword clue is the 7 letter word FRAILTY
I've seen this crossword clue in Financial Times
F
R
A
I
L
T
Y
More clues leading to the result FRAILTY
Buddy explains!

The straight part
We know that "weakness" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case weakness = frailty.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does FRAILTY mean?
FRAILTY noun- The condition quality of being frail, physically, mentally, or morally; weakness of resolution; liability to be deceived or seduced.
- A fault proceeding from weakness; foible; sin of infirmity.