I have a class library in C# has has several classes and a app.config. I have put ConnectionString in app.config as follows:
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As others have said, the library will use the config file for the executing application.
I'd like to add one suggestion though. One option to consider, for (in my opinion) a bit of polish and a professional touch is creating a custom configuration section. That would allow you to group the settings for your library and not leave any confusion about which settings are used by your library: http://msdn.microsoft.com/en-us/library/2tw134k3(v=vs.100).aspx