I think the tendency is to view running a half marathon the same way we view running a 5 or 10k even - that it is something you can do without stopping, more so to the point, it is something you SHOULD do without stopping.
For beginners and even some not so beginners running a half marathon without stopping is an unrealistic goal, and running a marathon without walking is as likely as Charlie Sheen returning to Earth. This is not your fault. This is not something that you should feel is a poor reflection on your abilities or inadequate training. Rather walking is something that you should put in your training.
If you look closely most half and full training programs for beginners work around 10 and 1's. Run 10 minutes and walk 1minute. You may think it will slow your time down but it will not, if anything you'll run faster overall and enjoy the process more. For longer distances pace groups* will have a more exact formula that they work with which is normally a variation on this.
So for now, keep training, and while you're counting up those miles work in 10 and 1's into your longer runs. If you don't have a watch, consider running 3 songs and walking until the chorus/middle of the next on your ipod, or when you map out your route figure out where you would be at the 10 minute mark and walk when you reach it....there are alternative ways of doing just about everything.
*Pace Groups: Not all races offer pacers, but a large majority do. I would highly suggest participating in this process. When I first started running half/full's I had this desire to just go out there and do it. The thing is if you struggle, pace yourself wrong etc your race can fall apart really quick. You may even think you're running a really good race and then you realize it's Mile 6 and you've messed up your pacing. I found in Houston that there is actually a distinct science to pace group. My race fell apart, more so my hips did and I was struggling when the 5:15 group found me. I continued with them for about 5 miles and realized that they actually had structured their race around the knowledge everyone was going to get tired, they built in time into their pace and they structured it, so even people like myself who ended up walking the last 4 miles in agony could finish in the legal time constraints. That may not be you and I pray it's never me again, but being able to talk to someone and participate in a little community for those 5 miles was the key determinant in my finishing.