Can I bypass bad onboard SATA with a PCI card?

My HP Pavilion (approx 3YO) starts bootup but after a couple of minutes it seems to forget that it has any CD or HDD.
If I power off and back on, the verbose startup sequence reporting doesn't report that it even sees the drives. After waiting for a while with the power off it shows in the bootup sequence again, and once again stops recognizing the drive after a while.

I've already blown out dust and checked connectors, other obvious stuff. The fans are turning.

I've had a similar problem a year ago while it was still under extended warranty, and they "replaced the motherboard", or at least that's what they said they did. Now, of course, the extended warranty has expired.

I bought a HDD thinking that this time it was the drive, but no dice -- in fact it stopped reading the DVD too which was when I noticed it didn't even list the drive in the verbose BIOS.

Since it seems to be the SATA controller that isn't working, is it possible to pop in a PCIE card, plug the drive(s) in there, and carry on? I'd like to find the lowest cost solution possible. The BIOS does have an option to disable the onboard SATA. Or am I on the wrong track entirely and there is something else to look at?