You must keep the MIT license notice intact, but you can license your software under whatever terms you want.
It is usually best to include a separate page in your documentation or in some built-in help system for all the open source libraries you are using. For example:
The following open source libraries are used by this application:
libexample -- http://example.com/
Copyright 2012 Jane Random Hacker
Permission is hereby granted ... <snip MIT license>
<copyright and license terms of another software>
Most web browsers like Firefox and Chrome have such pages, reachable from their “about this program” menu entry.
Such separate pages make it clear that the MIT license only applies to that component, not to the entire software.
In your source code the difference would usually be unambiguous, because third party components are usually kept in a separate directory from your own source code – and if you're using a package manager, your code doesn't include the dependency at all.