UPDATE: Looks like this is not possible. This isn't really a bug with the Map itself, but it kind of is. The Map uses a Canvas inside the Grid "RootElement". This Canvas holds the images for the map. When using a Canvas, it doesn't respect the bounds it's been given. You can reproduce the bug with the following XAML
<Border BorderBrush="Red" BorderThickness="2" CornerRadius="25">
<Grid>
<Grid>
<Canvas>
<Image Source="/Images/MyPicture.png"/>
</Canvas>
</Grid>
</Grid>
</Border>
The best approach is going to be to set an explicit style for the map. With this style any map that is used will have the rounded corners
<Style TargetType="esri:Map">
<Setter Property="Template">
<Setter.Value>
<ControlTemplate TargetType="esri:Map">
<Border CornerRadius="25" BorderBrush="{TemplateBinding BorderBrush}" BorderThickness="{TemplateBinding BorderThickness}" Background="{TemplateBinding Background}">
<Grid>
<Grid x:Name="RootElement" Height="Auto" Width="Auto"/>
<Rectangle x:Name="ZoomBox" Fill="#55FFFFFF" Stroke="Red" StrokeThickness="2" Visibility="Collapsed"/>
</Grid>
</Border>
</ControlTemplate>
</Setter.Value>
</Setter>
</Style>