Personally, if it were mine, I would use
http://www.acronis.com/en-us/personal/pc-backup/
They still are offering a free trial which, while limited in time, I believe contains all the features you would need for this one time project.
Create a system image of the entire drive and tell it to ignore bad sectors, which I believe, is under "error handling".
Then perform a restore, booting from the Acronis Rescue Disk that you can create from within the software and do a restore to your new drive, again tell it to ignore bad sectors (just in case).