In fact, it is impossible to integrate the new Time Capsule to an existing network. Why this is so and how to fix the problem, I will now describe in more detail as I possibly can in order to help others. The problem is not with the Time Capsule, but solely with the AirPort Utility. After intense internet research, we came across the following. Starting with version 6 of AirPort Utility, there are apparently a bug due to which only a new network can be created. There are indeed all options displayed (new network and integrate existing equipment replaced), but if you choose Option 2, you will be "Next" automatically redirected to the dialogue of Option 1. The current version of the utility is 6.3.2 and Apple still has not corrected this error. Thus, in order to integrate the Time Capsule to an existing network, you must you have an older version (5.6) get AirPort Utility.
Here arises an additional problem: Version 5.6 does not run on Mavericks (whether this is also true for Mountain / Lion, is unknown to me). To 5.6. to bring under Mavericks and running, please read the following tutorial [...]
With 5.6 to configure the Time Capsule is a breeze. She is quiet, elegant, runs without problems and does what I bought it. But it took about 3 hours until we have found the solution described above and there is a massive star deduction.