Absolutely possible.  You need to configure your routes properly to do this.  You want your __default route__ to go through your __eth1__, so your routing table should look like this:

    $ /sbin/route
    Kernel IP routing table
    Destination     Gateway         Genmask         Flags Metric Ref    Use Iface
    10.10.19.0      *               255.255.255.0   U     0      0        0 eth0
    192.168.1.0     *               255.255.255.0   U     0      0        0 eth1
    default         192.168.1.1     0.0.0.0         UG    0      0        0 eth1

Windows will look somewhat similar (with formatting variations of course) using the `route print` command.

You can set up the routes dynamically with the `route` commands on either platform.  I'm not sure what configuration options you need to set one as default (and the other as not-default, obviously)... will edit with that info.