How to Bind To Data within a Datatemplate of a ContentControl

I have the following simplified Example:

    <Window x:Class="TemplateBinding.MainWindow"
            xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
            xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"
            Title="MainWindow" Height="350" Width="525">
        <Window.Resources>
            <ResourceDictionary>
                <ResourceDictionary.MergedDictionaries>
                    <ResourceDictionary
                            Source="pack://application:,,,/TemplateBinding;component/PersonTemplate.xaml" />
                </ResourceDictionary.MergedDictionaries>
            </ResourceDictionary>
        </Window.Resources>
        <Grid>
            <ContentControl ContentTemplate="{StaticResource PersonTemplate}" />
        </Grid>
    </Window>

With:

    <ResourceDictionary xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
                        xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml">

        <DataTemplate x:Key="PersonTemplate">
            <Border Width="100" Height="100" Background="RosyBrown">
                <TextBlock Text="{Binding Path=FirstName}" VerticalAlignment="Center" TextAlignment="Center"/>
            </Border>
        </DataTemplate>

    </ResourceDictionary>

as my DataTemplate in a separate ResourceDictionary file.

I set my DataContext in the Construcor of my MainWindow and have verified it by just displaying the first name like this: <ContentControl Grid.Row="1" Content="{Binding FirstName}"/>.

In an other scenario wher I use a DataTemplate with a ListBox I do the Binding exactly the same way in my DataTemplate and it just works.

I know that the DataTemplate is working except the binding because it correctly shows the size and ba