TextBox expanding with surrounding Grid but not with text - wpf

Question

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>

Answer 1

You could use an invisible border (its hacky but it works - its how I tend to sort out dynamic textbox sizes in Xaml):
<Border BorderThickness="0" x:Name="border" Grid.Column="1" Margin="0.5" />
<TextBox Grid.Column="1" AcceptsReturn="True" TextWrapping="Wrap" Width="{Binding ActualWidth, ElementName=border}" Height="{Binding ActualHeight, ElementName=border}" />

Answer 2

Have you tried setting the MaxWidth property on just the TextBox?
Edit after OP's comment
I would try getting rid of the ScrollViewer. The sizing used in the Grid's layout should take care of re-sizing and the scroll bar settings on the TextBox should take care of the rest.

Answer 3

The answer is based on Leom's answer.
The solution works great when you enlarge the window, but the resizing is not smooth when you make the window smaller. As the textbox participates in the grid's layout, it has to perform layout process multiple times. You can fix that by putting the texbox in the canvas, so the change of the size of the textbox no longer triggers the grid's re-layout.
The updated code:
<Border BorderThickness="0" x:Name="border" Grid.Column="1" Margin="0.5" />
<Canvas Grid.Column="1">
<TextBox AcceptsReturn="True" TextWrapping="Wrap" Width="{Binding ActualWidth, ElementName=border}" Height="{Binding ActualHeight, ElementName=border}" />
</Canvas>