can I install Leopard 10.5 on my mini-mac 1.66Gz, intel Duo Core, 512 MB, 667 Hz

Hello & a warm welcome! 🙂

Yes you can, but 10.6 would be cheaper...

Leopard requirements/10.5.x...

* Mac computer with an Intel, PowerPC G5, or PowerPC G4 (867MHz or faster) processor

minimum system requirements

* 512MB of memory (I say 1.5GB for PPC at least, 2-3GB minimum for IntelMacs)

* DVD drive for installation

* 9GB of available disk space (I say 30GB at least)

Classic/OS9 Apps no longer supported.

May be available from Apple again, or not...

https://discussions.apple.com/thread/4720126?tstart=0

Trouble is Apple no longer sells it, check eBay & such for the Retail version, not the Gray Discs...

http://www.ebay.com/sch/i.html?_nkw=mac+os+x+leopard+retail+10.5

Snow Leopard/10.6.x Requirements...

General requirements

* Mac computer with an Intel processor

* 1GB of memory (I say 4GB at least, more if you can afford it)

* 5GB of available disk space (I say 30GB at least)

* DVD drive for installation

* Some features require a compatible Internet service provider; fees may apply.

* Some features require Apple’s MobileMe service; fees and terms apply.

Buy Snow Leopard > http://store.apple.com/us/product/MC573/mac-os-x-106-snow-leopard

Call Apple Sales...in the US: 1-800-MY-APPLE. Or Support... 1-800-275-2273

Other countries...

http://support.apple.com/kb/HE57

It looks like they might still have it...

http://store.apple.com/us/product/MC573Z/A?fnode=MTY1NDAzOA

Yah, go for it.

Oh, forgot, I'd recommend upgrading to 2 GB of RAM regardless which OSX you decide on...

http://eshop.macsales.com/MyOWC/Upgrades.cfm?model=252

Installation videos...

http://eshop.macsales.com/installvideos/mac_mini_06-08_mem_hd_opt/