Wpf – TextBox expanding with surrounding Grid but not with text

A window has a Grid with two columns. The left column contains a control with a constant width but with a height that adapts. The right column contains a TextBox that takes up all remaining space in the Grid (and thereby in the Window).

The Grid is given a minimal width and height and is wrapped within a ScrollViewer. If the user resizes the window to be smaller than the minimal width/height of the Grid, scrollbars are displayed.

This is exactly how I want it to be. However, a problem occurs when the user starts typing text. If the text is to long to fit in one line in the TextBox, I want the text to wrap. Therefore I set TextWrapping="Wrap" on the TextBox. But since the TextBox has an automatic width and is wrapped in a ScrollViewer (its actually the whole Grid that is wrapped), the TextBox just keeps expanding to the right.

I do want the TextBox to expand if the window is expanded, but I don't want the TextBox to expand by the text. Rather the text should wrap inside the available TextBox. If the text don't fit within the TextBox height, a scrollbar should be displayed within the TextBox.

Is there a way to accomplish this?

Below is some code that shows my problem:

<Window x:Class="AdaptingTextBoxes.MainWindow"
    xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
    xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"
    Title="MainWindow" Height="300" Width="400" Background="DarkCyan">
<Grid Margin="10" Name="LayoutRoot">
    <ScrollViewer HorizontalScrollBarVisibility="Auto" VerticalScrollBarVisibility="Auto">
        <Grid MinWidth="300" MinHeight="200">
            <Grid.ColumnDefinitions>
                <ColumnDefinition Width="auto" />
                <ColumnDefinition Width="*" />
            </Grid.ColumnDefinitions>

            <Button Grid.Column="0" Margin="0,0,10,0" Content="Button" Width="100" />

            <TextBox Grid.Column="1" AcceptsReturn="True" TextWrapping="Wrap" ScrollViewer.HorizontalScrollBarVisibility="Disabled" ScrollViewer.VerticalScrollBarVisibility="Auto" />
        </Grid>
    </ScrollViewer>
</Grid>
</Window>