Sorry, take out Dir & "\" &.
Code:
        Dim Dir As String = "C:\Directory"
        Dim fileList As New List(Of String)
        fileList.AddRange(My.Computer.FileSystem.GetFiles(Dir))

        For x As Integer = 0 To fileList.Count - 1
            Dim theFile As String = fileList.Item(x)

            If My.Computer.FileSystem.GetFileInfo(theFile).Extension = ".dem1" Then
                My.Computer.FileSystem.DeleteFile(theFile)
            End If
        Next