Read from App.config in a Class Library project

As stated in my comment, add the App.Config file to the main solution and not in the class library project.

You dont need to add app.config file. If you create class library for web based application then you can fetch connection string directly from web.config file

OR

You can add any text file with connection string in it and fetch that string . using this

public static ConnectionStringSettings ConnSettings
{
    get
    {
        string connectionStringKey = null;
        connectionStringKey = ConfigurationManager.AppSettings.Get("DefaultConnectionString");
        return ConfigurationManager.ConnectionStrings[connectionStringKey];          
    }
}