I have manged to solve the issue with Apple's own USB Ethernet Adapter Model A1277:
Looking in System Information, I found out it has the Product ID 0x1402.
With that I found the source code of a driver mentioning various ASIX models:
http://lists.denx.de/pipermail/u-boot/2011-May/093617.html
It turns out that A1277 corresponds to ASIX_MODEL_AX88772
The drivers for these you will find here:
http://www.asix.com.tw/download.php?sub=driverdetail&PItemID=136
I downloaded "Apple Mac OS X 10.9 Driver" and it worked.
It boggles the mind that a totally functional Apple product bought in 2010 is unsupported in 2013, while drivers for these still exist and are maintained.