You would need a custom ROM that supports moving apps to the Ext3 partition. I would recommend searching the development section of xda-developers forums for your handset for such a ROM.
If a ROM does not support apps2ext natively, you can try your luck with a script (example for HTC Dream/Magic here) that can be installed on top of another ROM.
[Edit]: just adding stuff from my comments to the answer.
If you are on Android version 2.2 and above, you can use the built-in "Apps-to-SD" functionality to move apps to SD Card. Go to
Menu -> Settings -> Applications -> Manage applications then tap on any app and under the "Storage" section tap the "Move to SD" button. After a few seconds you will see the button's name change to "Move to Phone" and size of the app decrease significantly.
This method is not without drawbacks. First off, an app needs to be explicitly coded to support this functionality. Second, widgets and apps that run as a service (e-mail, virtual keyboards, weather notifications, etc.) don't work when moved to SD, and must reside on the phone's internal memory. Furthermore, not the entire app moves to the SD Card. A part of it stays on the phone, along with its cache and data/settings. If you have a phone with a small amount of internal app storage, using the built-in Apps-to-SD will just prolong the inevitable "out of memory" errors.
The Apps2ext scripts resolve most of these shortcomings. It "fools" the phone into thinking that the Ext2/3/4 partition is part of the phone's internal memory. This means that widgets and services will work just fine with this method. Also, since you can make your partition as large as you wish (recommended not to exceed 2GB), the space issues disappear. The scripts also provide options for moving caches and data to the Ext partition in order to free up even more internal memory.
The drawback of the Apps2ext method is that it's well, a hack. First off, your phone needs to be rooted. Second, you will no longer be able to un-mount and remove the SD Card from the phone, since ALL of the apps (including system ones) reside on the Ext partition. Third, you are putting a lot more strain on the SD Card from increased writing, which may shorten its lifespan. Finally, as with any hack it may make your phone less stable.