VB.NET Connection string (Web.Config, App.Config)

Not clear where My_ConnectionString is coming from in your example, but try this

System.Configuration.ConfigurationManager.ConnectionStrings("My_ConnectionString").ConnectionString

like this

Dim DBConnection As New SqlConnection(System.Configuration.ConfigurationManager.ConnectionStrings("My_ConnectionString").ConnectionString)

If it's a .mdf database and the connection string was saved when it was created, you should be able to access it via:

    Dim cn As SqlConnection = New SqlConnection(My.Settings.DatabaseNameConnectionString)

Hope that helps someone.