How can I prioritize WPF textbox wrap over autosize?

I have a number of situations where I have panels or grids that resize automatically, but if they contain a TextBox with TextWrapping="Wrap", the TextBox continues to expand the panel/grid to the right long before it really needs to, such as the image below:

What I am looking to do is to have the TextBox fill its area by wrapping text before it tries to expand to the right. A simplified example of the issue is:

<Grid>
    <Grid Background="Black" />
    <Grid VerticalAlignment="Top" HorizontalAlignment="Left" Background="White">
        <Grid.ColumnDefinitions>
            <ColumnDefinition Width="Auto"></ColumnDefinition>
        </Grid.ColumnDefinitions>
        <TextBox TextWrapping="Wrap" Height="120" MinWidth="200" />
    </Grid>
</Grid>

I found a similar question on Stack Overflow here, but the best solution posted did not allow the TextBox to expand. That solution was something like:

<Grid>
    <Grid Background="Black">
    </Grid>
    <Grid VerticalAlignment="Top" HorizontalAlignment="Left" Background="White">
        <Grid.ColumnDefinitions>
            <ColumnDefinition Width="Auto"></ColumnDefinition>
        </Grid.ColumnDefinitions>
        <Border BorderThickness="0" x:Name="border" Margin="0.5" />
        <TextBox TextWrapping="Wrap" Height="120" MinWidth="200" Width="{Binding ActualWidth, ElementName=border}" />
    </Grid>
</Grid>

Any ideas other than extending TextBox with modified behaviors?