Non breaking space in XAML vs. code

This works fine, and correctly inserts non-breaking spaces into the string:

<TextBlock Text="Non&#160;Breaking&#160;Text&#160;Here"></TextBlock>

But what I really need is to replace spaces with non-breaking spaces during data binding. So I wrote a simple value converter that replaces spaces with " ". It does indeed replace spaces with " " but " " is displayed literally instead of showing as a non-breaking space. This is my converter:

public class SpaceToNbspConverter : IValueConverter
{
    #region IValueConverter Members

    public object Convert(object value, Type targetType, object parameter, CultureInfo culture)
    {
        return value.ToString().Replace(" ", " ");
    }

    public object ConvertBack(object value, Type targetType, object parameter, CultureInfo culture)
    {
        throw new NotImplementedException();
    }

    #endregion
}

Does anybody know why it works in XAML, but not in code?