Sheikh Hamad al-Haydarî, professor at al-Imâm University
Putting this cat to death is not permissible. If you are afraid of it causing harm or are worried on account of the damage it may be doing in your house, then you may take it away to a remote area where it will be capable of surviving.
As for the cat being weak and unable to survive outside the home, this is not an excuse for you to kill it. Send it to any place where it will be able to find its sustenance. In this case, if the cat dies after that you will not be sinful.
The Prophet (peace be upon him) said: “A woman was punished on account of a cat she had kept imprisoned until it died, and because of it she entered the Fire. She neither provided it with food and drink nor freed it so it could feed upon the vermin on the ground.” [Sahîh al-Bukhârî and Sahîh Muslim]
What concerns us are the words: “…nor freed it so it could feed upon the vermin on the ground.”
If you set this cat free in some suitable place where it can look for its food, then you will not be sinful if it dies.
And Allah knows best.
Source: Islam Today