Removing IsEnabled and setting the TextBox as ReadOnly would allow you to select the text but stop user input.
IsReadOnly="True"
The only issue with this approach is that although you won't be able to type in the TextBox it will still look 'Enabled'.
To get round that (if you want to?) you can just add a style to lighten the text and darken the background (to make it look disabled).
I've added the following example with a style that will flick the textbox between a disabled and enabled look.
<Window x:Class="WpfApplication1.Window1"
xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"
Title="Window1" Height="300" Width="300">
<Window.Resources>
<Style TargetType="{x:Type TextBox}">
<Style.Triggers>
<Trigger Property="IsReadOnly" Value="True">
<Setter Property="Background" Value="LightGray" />
</Trigger>
<Trigger Property="IsReadOnly" Value="True">
<Setter Property="Foreground" Value="DarkGray" />
</Trigger>
<Trigger Property="IsReadOnly" Value="False">
<Setter Property="Background" Value="White" />
</Trigger>
<Trigger Property="IsReadOnly" Value="False">
<Setter Property="Foreground" Value="Black" />
</Trigger>
</Style.Triggers>
</Style>
</Window.Resources>
<Grid>
<TextBox Height="23" Margin="25,22,133,0" IsReadOnly="True" Text="monkey" Name="textBox1" VerticalAlignment="Top" />
<Button Height="23" Margin="25,51,133,0" Name="button1" VerticalAlignment="Top" Click="button1_Click">Button</Button>
</Grid>
private void button1_Click(object sender, RoutedEventArgs e)
{
textBox1.IsReadOnly = !textBox1.IsReadOnly;
}