The modem is fine, it's just not designed to work with a router without some re-configuration.
Connect the modem directly to your computer using an Ethernet cable
Pull up your browser and enter 192.168.1.254 into the address bar and press enter. That should pull up the modem configuration pages
Look for an Advanced tab on the left. You may need to enter the modem access code when you do this. It is on the bottom of the modem
Click the PPP Location Tab
Change this setting to "PPP is on the computer (or router)..... or, if you do not see this setting, there should be a setting for Bridge Mode.
Make your selection and click Save or similar and modem will restart with the new settings
Important....if you need to get back to the original modem settings again, you will need to first hold the reset button in on the back of the modem for 10 seconds, then let it restart. That will get it back to original factory settings if you need to get back to those settings.
With the AirPort Express powered up, hold in the reset button on the device for 10-12 seccnds and then release the reset button. Allow 40-45 seconds for the Express to restart at which time it will slowly blink amber
Connect the Ethernet cable from the modem to the AirPort Express
Make sure you are logged onto the wireless signal for the AirPort Express. It provides a temporary network with a name like Apple Network xxxxxx. Log on to that network. There is no password.
Open AirPort Utility - Click Manual Setup
Click the Base Station tab/label located just under the row of icons to assign a name to the Express, a device password (write that down) and adjust Time Zone settings
Click the Wireless tab next to the Base Station tab
Wireless Mode = Create a wireless network
Wireless Network Name = Your choice
Check mark next to Allow network to be extended
Radio Mode = 802.11n 802.11b/g compatible
Channel = Automatic
Security = WPA2 Personal
Password = Your choice
Confirm Password
Click the Internet icon
Connect Using = PPPoE
User Name = probably your email address with your provider
Password = password with your provider
Verify Password
Service Name....Leave this completely blank
Connection = Always On
Disconnect if Idle = Never
Click Update at the lower right and allow 40-45 seconds to restart. You may see a message about "setup over WAN". Click to ignore.
As soon as you see the message in AirPort Utility that the settings have been stored...Close AirPort Utility.
Wait for a green light on the AirPort Express and then click the fan shaped AirPort icon at the top of the computer screen to look for the new network that you just named. Log on and enter the password.
You should have an Internet connection.
If not, power everything down. Start the modem first, then the Express, then your computers