You have to be more specific with your phone. The Samsung III does not let us know which phone you have.
There is the Samsung Galaxy S III and the Samsung Galaxy Note III. From the description of your problem, I would suspect you have the Samsung Galaxy Note III.
You can go into your settings under wifi and disable the auto switch if your wifi signal is not strong enough. As I don't have the Note III in front of me right now, I don't know the exact wording of the setting. I am confident you will be able to select the correct checkbox.
Settings > Wifi > Menu > Advanced > Turn off Auto Network Switch
Of course, on the Samsung Galaxy Note III, this problem is not with your wifi signal, but with the phone.
The above setting will simply not let your phone switch back to cellular data when it determines your wifi signal is unusable. You will simply stay on wifi without a connection until your phone cycles back through releasing the wifi signal and reacquiring it. Until it reacquires the signal, you will simply be without a data connection.