Since, effectively, you will have two routers in series, you will want to reconfigure the Time Capsule (TC) as a bridge. This will allow the Verizon router to continue to provide both NAT & DHCP services for both wired & wireless clients regardless of which of the two routers they connect to.
To reconfigure the TC as a bridge you will need to use the AirPort Utility: AirPort Utility > Select the TC > Manual Setup > Internet > Internet Connection tab > Connection Sharing = Off (Bridge Mode)
Thank you Neturonboy for asking this question and thank you Tesserax for answering it. It's amazing that Verion's site has nothing about this; they do mention the Time Capsule, but nothing about this typical configuration. I have the GT784WNV model DSL modem/router, which is probably similar to yours. As soon as I turned Connection Sharring Off (Bridge Bode) and updated changes it worked.
However, I couldn't find where to do this at first using Mac OS X Lion. Luckily, I had an older laptop, which had the more familiar looking AirPort Utility version 5.4.2 you're talking about. After I had switched to Bridge Mode, I went back to Lion's version and could easily see where this setting is located. In Lion's (10.7.4) AirPort Utility, click on the image of your Time Capsule (there is no "Manual Setup" button in Lion), then select Network (not Internet), then change Router Mode (not Connection Sharing): to Off (Bridge Mode).
About Guest Networks:
Note, if you have a Guest Network, then once you go to Bridge Mode, this feature is turned off; the whole Guest Network tab disappears from the choices under Airport (using the older AirPort Utility). Since these Verizon modem/routers are also wireless, your guests can access the network directly from Verizon's router instead of going through your Time Capsule. Perhaps having a wireless router feature for your DSL modem makes the Time Capsule's Guest Network option redundant. You can get the network name, and other security information from the label on Verizon's router, but there is also a WPS (Wi-Fi Protected Setup) feature that looks like it might be useful for sharing your network. There's a physical button on the modem and there's a settings page on the routers' web browser accessible settings at 192.168.1.1.. I'm still looking into this.