I'm totally comfy on my comfort bike. I ride on calm roads during not busy times. I'm one of those nine month pregnant people that still ride. For me, I'm not worried about balance, but I've always ridden. My center of gravity feels so low on the comfort bike that I could not imagine falling off. I'm sure it's possible, but to me, it doesn't feel likely. But, you should only due what you feel comfortable with. If you're worried about falling that probably won't help you ride comfortably. As someone else said on this post, the danger is much higher of being hit by a car than falling off a bike.
I've enjoyed biking throughout my pregnancy despite most people telling me that I shouldn't ride because I might fall. I won't take advice easily from people who don't exercise. Biking is more comfortable than the running I did most of my pregnancy although now, I am swimming and not biking or running because it has been over 90 degrees for more than 10 days now.
I would say, like the others here, you should do what you feel comfortable with. It doesn't matter if you find some article that says your chances of falling off are 2% during any given ride. What will that information do for you? Also, if you do decide to bike, as with any exercise while pregnant, you have to be more cautious to not push yourself too hard. Exercising is tougher being pregnant. Know yourself.
For example- yes, I can run 10 miles. But I kept it to a slow 3 miles at most while pregnant. Sometimes, I just did one. Some people they would tell me, books say you shouldn't run more than 2 miles. It's a book. It's not my body. The people talking shouldn't run 2 miles because they would collapse.
I've rambled enough. Know yourself and do only what you feel comfortable with. I would totally encourage you to exercise somehow though. I'm due next week and I've enjoyed a great pregnancy and even now I have energy and feel like I'm headed into labor in excellent condition. I happy to have chosen early on to make it a priority to exercise. It is definitely paying off. I hope you find a routine that works for you that you can stick with.