Go to "Network" in settings. Then click on your "Built In Ethernet" device on the left if it isn't already selected. Click on "Advanced" near the bottom right corner of the pane. Next, click on the "Ethernet" tab that comes up (far right). Finally, select "1000baseT" in the "Speed" drop down and click OK. This will force OSX to use a Gigabit connection rather than auto-select it (which sometimes doesn't pick the right speed here for some unknown reason). Keep in mind that if you connect it to a 100baseT connection/hub/switch/whatever, it will not work in this mode (i.e. 1000baseT requires a full gigabit connection all the way to the router, so use gigabit switches, etc. where appropriate).
If by AEBS you mean the Airport Express Base Station, it won't work because they are 100baseT. Just connect the Cat6 cable to a Gigabit switch on the far end instead (they don't cost that much; I use one here in the same room as my router to give me more connections) of the AEBS (you don't need to create a new network; the switch will extend the primary one from the Time Capsule and maintain the Gigabit speeds).
