Thanks Walter, this was exactly what I needed. The style using a
relativesource binding is something I couldn't really get a grasp on, this is
perfect (though not immediately intuitive).

For others who are intersted here is the datatemplate that solves this issue:
<DataTemplate x:Key="itemTemplate">
     <Grid>
       <Grid.Resources>
         <!-- This style is applied to the StackPanel which contains the
controls only
              displayed by the ListBox's selected item. -->
         <Style x:Key="selectedPanelStyle">
           <Style.Triggers>
             <DataTrigger
               Binding="{Binding
                         RelativeSource=
                         {
                            RelativeSource
                            Mode=FindAncestor,
                            AncestorType={x:Type ListBoxItem}
                         },
                         Path=IsSelected
                        }"
               Value="False">
               <Setter Property="StackPanel.Visibility" Value="Collapsed" />
             </DataTrigger>
           </Style.Triggers>
         </Style>
       </Grid.Resources>
       
       <Grid.RowDefinitions>
         <RowDefinition />
         <RowDefinition />
       </Grid.RowDefinitions>
       
       <!-- This is displayed whether the ListBoxItem is selected or not. -->
       <TextBlock Text="{Binding Path=Name}"/>
       
       <!-- This is only displayed when the ListBoxItem is selected. -->
       <StackPanel Grid.Row="1" Style="{StaticResource selectedPanelStyle}">
         <Button>Edit Item...</Button>
       </StackPanel>
     </Grid>
   </DataTemplate>

Note the: Binding="{Binding
                         RelativeSource=
                         {
                            RelativeSource
                            Mode=FindAncestor,
                            AncestorType={x:Type ListBoxItem}
                         },
                         Path=IsSelected
                        }"

That is what allows you to know whether or not the item is selected.