Move external HD to a new drive, but have Time Machine think it's the same?

I have two external drives, one for data and one for Time Machine backups. Time Machine backs up both my internal HD and my external HD. My external HD is dying, so I bought a new one and plan to migrate the data over.

Is it possible to somehow move the data to the new HD and still have Time Machine think that the new HD is the old one, since the data is the same? I usually just use Finder to move files from the old HD to new one (and then remove the old HD and rename the new one to use the same name), but from what I understand that wouldn't copy over the Time Machine data, however.