Hardware disk duplicators tend to be very expensive (hundreds to thousands of USD).
Multi-device adapters (i.e. a device which can be hooked up to your PC that supports both PATA and SATA disks) tend to be USB 2.0-based, so transfer speed is not that great (around 30-35MBytes/second). And in either case, sometimes the cabling/wiring used in these adapters tends to be sub-par (inducing CRC errors, etc.). Many of these also use USB-SATA bridges that do not permit SMART passthrough (meaning you can't view the SMART attributes for an attached drive). I will admit I've used these adapters with success in the past (for transferring from a disk to a PC and back), but YMMV.
I don't really have any recommendations otherwise. Try to remember: this is a one-time situation (migrating from PATA to SATA). A little bit of pain only once (dealing with the internals of your case, etc.) isn't that bad. It'd be different if you had to do this every day for a few years, or had a thousand PATA disks. --
Making life hard for others since 1977.
I speak for myself and not my employer/affiliates of my employer.