Thanks to a comment posted by Andrew to my Revit Tip - Masking Regions post, I am a step further along on my Revit journey. He correctly pointed out that the Linework tool would be a better choice for getting rid of the unwanted line.
On the Modify ribbon tab, on the View panel, click on the Linework tool, shown below in Revit Architecture 2011.(As always, click on any reduced-size image to see the image full size. Use the Back button on your browser to return here.)
On the Modify|Linework ribbon tab, on the Line Style Panel, set the Line Style to <Invisible lines> in the Line Style drop down list.Now you can select the unwanted linework in the View in which you do not want to see it, and make it invisible with a single click. That is far more efficient than using a Masking Region to hide a single line.
If you change your mind later, you can restore the linework you made invisible by using the same Linework tool and setting the Line Style to <By Category> (or any other visible Line Style, if you do not want to use its default appearance).