I am not sure if this HD is compatible with my computer, so am wondering if anyone would
be kind enough to inform me. The HD I am looking to buy can be found:
https://www.cclonline.com/product-info.asp?id=19278
My laptop is a Sony Vaio VGN A217M, and the broken old HD was a Hitachi IC25N080ATMR04.
Thanks so much.

This drive is fine as a replacement. The IC25 is a 2.5" ATA, so it will be compatible.

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Data Recovery Ireland

As long as the laptop supports 48-bit lba, then yes

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You don't have to backup all of your files, just the ones you want to keep.

I don't know if this will work, but in the event that the laptop's BIOS is limited to 28-bit LBA, then create a 128GB partition inside that limit. Install your OS on that partition. After the OS boots, its own 48-bit LBA drivers should assume control, and you can then partition and format the remaining disc space.

This is true but will lead to problems down the line

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You don't have to backup all of your files, just the ones you want to keep.