I would like to download some favorite pieces of '70's and '80's
Christian music.  I am willing to pay a few dollars for each song, but
I don't want to pay a membership fee or buy the whole album just to
get the song I like.  Are there any services on the web that support
this?

Dear Statman, 

The following sites offer downloads of Christian music: 

Share a Song 
http://www.sharesong.org/worship_music.htm
You could search by artist, song title and other features. 

Praise Song 
http://www.praisesong.net/Freepraisemusic.htm 

Emmanuel's site
http://www.emmanuel.kiev.ua/music_groupE.html 
I doubt if you'll find what' you're looking for here, since most of
the site contains contemporary Christian artists.

Gospel Crib
http://gospelcrib.8m.com/ 

Bread Site
http://breadsite.org/music.htm 

Ekhardt
http://ekhardt.com/music/ 
Only Ekhardt music. 

Christian Music Exchange 
http://users.aol.com/cmx/ 
Exchange of Christian music, including also songs or singles. 


In addition, some of the song-download sites could be used to locate
and download Chritian songs:

Napster (requires registration, but you pay for songs, not for
membership): <http://www.napster.com/>

iTunes <http://www.apple.com/itunes/download/> - each song costs $0.99

Further sites could be found at : 
http://directory.google.com/Top/Society/Religion_and_Spirituality/Christianity/Music/Sound_Files/MP3/

I hope that helps. Please contact me if you need any further
clarification on this answer before you tip/rate it.

Merry Christmas!

Very thorough answer.

Thank you for the rating and the tip!