Program to read good files off of failing hard disk

Is there a low cost program that will read the good files off of my
failing hard disk and write them to my new hard disk? All the ones I
have found check the source drive and if there is one bad sector in one
of the thousands of files they refuse to copy anything. Thank you in
advance for all replies.
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Sometimes when I play with my cat I get him to go around in a circle really
fast until he gets so dizzy that he can't stand up anymore. Then I get him
to go around in the opposite direction. I call it winding the cat. I think
I wind him too much because the clock in his stomach runs fast.