This procedure concerns the following software versions:

ruby 1.8.5 (2006-08-25) [i386-cygwin] sqlite version 3.3.8 sqlite3-ruby (1.1.0) (Ruby gem) rails (1.1.6) (Ruby gem)

Install Cygwin and Rails as described on RailsonWindowsCygwin

Then, download the sqlite3 sources from the SQLite download page. I compiled the 3.3.8 version from
the source tarball.

Extract the tarball and read the README for compilation instructions.
You will need to install GCC and Make using the Cygwin installer.

After compilation you will have sqlite3.exe. You will also need to build
the .dll. In the sqlite source directory type

make sqlite3.dll

Then copy the .dll and the .exe to somewhere on your path, eg:

cp sqlite3.exe sqlite3.dll ~/bin

Now install the sqlite3-ruby gem (choosing the latest mswin32 version)

gem install sqlite3-ruby

You have to make a change to this file

c:/cygwin/lib/ruby/gems/1.8/gems/sqlite3-ruby-1.1.0-mswin32/lib/sqlite3/driver/dl/api.rb

So that it reads

when /win32/, /i386-cygwin/
  "sqlite3.dll"

Cygwin still tried to load in the sqlite3_api.so file so you may need to delete it

rm /usr/lib/ruby/gems/1.8/gems/sqlite3-ruby-1.1.0-mswin32/lib/sqlite3_api.so

Now you should be able to specify sqlite3 as the database in your
database.yml, for example :

development:
  adapter: sqlite3
  database: db/residence_development.db