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.

**Edit**:  If you're working with the GNOME or KDE GUI network managers, look for a "set this interface as default" option in the configuration for your `eth1` device.  

If you're configuring `/etc/network/interfaces` by hand, take a look at [the examples in this HOWTO][1].  In particular, the `up` option allows you to run commands after an interface comes up.  In your case, you may need to use that to run a route-delete command on an extra default route, or to run a route-add if neither of your interfaces set themselves as a default route:

    # example /etc/network/interfaces
    # replace the IP addresses in the route-del and route-add commands below
    # with those appropriate to your network
    
    auto eth0
    iface eth0 inet dhcp
        up route del default gw 10.10.19.1
        # runs a route-delete if dhcp adds a default gateway for this interface
    
    auto eth1
    iface eth1 inet dhcp
        up route add default gw 192.168.1.1
        # runs a route-add if dhcp neglects to add a default gateway for this interface

  [1]: http://www.cyberciti.biz/faq/setting-up-an-network-interfaces-file/