This is a common problem (and I wish I had read it sooner) for all development, not just ASP.NET. Being one of its developers, my team naturally uses BuildMaster internally for the entire release process, and for most scenarios it's free. Within the tool, we are able to perform all the standard CI builds to create artifacts and then set up an automation process to deploy these artifacts to any one of the 40+ servers we have internally or externally hosted, depending on the specific application or environment.
Since you specifically mentioned deployment to different testing environments, this is a fundamental aspect of the tool. The idea is to model the environment workflow (e.g. Integration -> QA -> Production) you already have in place and essentially promote a build all the way from source control to production. Most times, it's as simple as adding a deployment action that deploys an artifact to the environment, other times it can be much more complex.
You also casually mentioned configuration file changes are part of deployment, which is another built-in component to BuildMaster. The idea we had was to use the tool itself as the central hub for all configuration files and deployments, thus ensuring the latest changes are applied automatically with a simple "deploy configuration files" action in your deployment plan.
One thing you didn't mention with regard to this process is the database deployment aspect. Most ASP.NET applications require an associated database, otherwise they could just be static HTML files. It is crucial that the database schema gets updated to the appropriate database version with every deployment. There is, not surprisingly, a module within BuildMaster that handles this for you as well. The idea is to store DDL-DML scripts within the tool itself, and by executing scripts only once per environment, it ensures that all of your databases across each environment are up-to-date as your builds are deployed through them. Other scripts (e.g. stored procedures, views, triggers, etc.) are essentially code files and therefore belong in source control. These DROP-CREATE-CONFIGURE type scripts can be run each and every time in most cases with a simple deployment action.
Another piece of the deployment puzzle that most developers do not think about is process automation. Many developers need to perform sign-offs or fill out change request forms in order to manually perform these processes. Again, this is all available as part of the automated workflow setup within BuildMaster. You can setup blockers that do not allow promotion to say the QA environment unless all unit tests have passed, or block promotion to the Staging environment unless someone from the QA team approves the build and all issues in your issue tracking tool are resolved/closed for that particular release.
While I realize I left out CC.NET from the answer, our applications are all built and deployed through BuildMaster so we no longer need it, though we could however just as easily pickup the artifacts from a drop location and deploy them in later environments.