Fixed! It turned out Sonar WAS listening to a second sound card - window's "High Definition Audio Device", which came with the machine, besides the ASUS card.  When I disabled it, Sonar let me know it was no longer available - and the stopping problem went away!  As Kolle suggested, it was due to another sound card interfering.
 
Also - I do have an "ASIO" ASUS driver (the 2011 7.12.8.1794 version), though I thought I didn't.  The driver mentions fixing an ASIO issue in the description.  Only the 2010 7.12.8.17731 version mentions ASIO in the name ("bit-perfect ASIO driver"), so it's not immediately clear that the later versions still have ASIO.  But the ASIO mode works fine in Sonar, so I guess they do.