What is the title of a science fiction movie made between the fifties
and sixties where the futuristic society employs androids as
companions who try to hide their identity, and at the end of the film
government scientists revel that they have transplanted the
consciousness of prominent citizens to an updated android model to
promote longer life?

Hello.

It sounds like you're describing "The Creation of the Humanoids" (1962).

"In the distant future, mankind has created a race of robots, who are
becoming more intelligent all the time. While the Flesh and Blood
League tries to remind the robots of their underclass place in
society, a scientist has developed a way to transfer human memories
into a robot body."
source:  RottenTomatoes.com:
http://www.rottentomatoes.com/m/creation_of_the_humanoids/about.php?rtp=1

Also see this review:
"The Creation of the Humanoids is the most sophisticated exploration
of intelligent robots in human society to hit the big screen until
Blade Runner (82).
Nuclear war has eliminated 92% of humanity. The humanoids, robots
developed from pre-war technologies, are mass-produced to help rebuild
urban centers and take up other slack...
The Order?s Captain Cragis (Don Megowan) uncovers a robot plot to seed
the population with the R-96 (just 4% off fully human)?humanoids
implanted with the memories of the recently dead, and practically
indistinguishable from people. He eventually discovers a Robot Central
Committee for the Preservation of Mankind?"
http://www.worldlyremains.com/Humanoids.html

Also see this newsgroup message:
http://groups.google.com/groups?hl=en&lr=&ie=UTF-8&c2coff=1&selm=3t91ph%24ltf%40apollo.it.luc.edu

Also see the Internet Movie Database entry:
http://www.imdb.com/title/tt0055872/

Videos on Ebay:
http://search.ebay.com/creation-humanoids_W0QQsokeywordredirectZ1QQfromZR8

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search strategy:
imdb.com, plot search: androids

I hope this is the movie that you had in mind. If not, please request
clarification. Thanks.

How did you find that information so quickly?