apphubフォーラムの「EricFleck-Microsoft」から助けを得ました。彼の解決策は次のとおりです。
<phone:PhoneApplicationPage
...
xmlns:c4f="clr-namespace:Coding4Fun.Phone.Controls;assembly=Coding4Fun.Phone.Controls">
<phone:PhoneApplicationPage.Resources>
<ControlTemplate x:Key="MsgPropmtNoBorder" TargetType="c4f:MessagePrompt">
<Grid VerticalAlignment="Stretch">
<Rectangle Fill="{StaticResource TransparentBrush}" />
<Border VerticalAlignment="Top"
Margin="10"
Background="{TemplateBinding Background}"
BorderThickness="0"
BorderBrush="{StaticResource PhoneForegroundBrush}">
<StackPanel Margin="10">
<TextBlock
Text="{TemplateBinding Title}"
Margin="0,-10,-25,10"
FontSize="30"
TextWrapping="Wrap" FontFamily="Segoe WP Light" />
<ContentPresenter Content="{TemplateBinding Body}" />
<StackPanel
Margin="0,10,0,0"
Name="actionButtonArea"
Orientation="Horizontal"
HorizontalAlignment="Center" />
</StackPanel>
</Border>
</Grid>
</ControlTemplate>
</phone:PhoneApplicationPage.Resources>
MessagePrompt prompt = new MessagePrompt();
prompt.Body = new WPUC();
prompt.ActionPopUpButtons.Clear();
prompt.Overlay = new SolidColorBrush(Color.FromArgb(155, 41, 41, 41));
prompt.Template = (ControlTemplate)this.Resources["MsgPropmtNoBorder"];
prompt.Show();
結果のポップアップは次のとおりです。