Sure, it's as simple as setting the Background and BorderBrush properties on the ListBox to Transparent and then setting a Background for the ListBoxItems:
<StackPanel Background="Red">
<ListBox Background="Transparent" BorderBrush="Transparent">
<ListBox.Resources>
<Style TargetType="{x:Type ListBoxItem}">
<Setter Property="Background" Value="White" />
<Setter Property="Margin" Value="1" />
</Style>
</ListBox.Resources>
<ListBoxItem Content="First Item"/>
<ListBoxItem Content="Secton Item"/>
</ListBox>
</StackPanel>
NOTE: I added a Margin to the ListBoxItems just to demostrate the spacing between the ListBoxItems will show all the way through to the surrounding StackPanel's red background.