The steps below worked for me. It uses the taps gem, created by Heroku and mentioned in Ryan Bates's Railscast #342. There are a few steps but it worked perfectly (even dates were correctly migrated), and it was far easier than the Oracle -> DB2 or SQL Server -> Oracle migrations I have done in the past. 
Note that SQLite does not have a user id or password, but the taps gem requires something. I just used the literals "user" and "password".
Create the Postgres database user for the new databases
$ createuser f3
Shall the new role be a superuser? (y/n) n
Shall the new role be allowed to create databases? (y/n) y
Shall the new role be allowed to create more new roles? (y/n) y
EDIT - Updated command below - use this instead
$ createuser f3 -d -s
Create the required databases
$ createdb -Of3 -Eutf8 f3_development
$ createdb -Of3 -Eutf8 f3_test
Update the Gemfile
gem 'sqlite3'
gem 'pg'
gem 'taps'
$ bundle
Update database.yml
#development:
#  adapter: sqlite3
#  database: db/development.sqlite3
#  pool: 5
#  timeout: 5000
development:
  adapter: postgresql
  encoding: unicode
  database: f3_development
  pool: 5
  username: f3
  password:
#test:
#  adapter: sqlite3
#  database: db/test.sqlite3
#  pool: 5
#  timeout: 5000
test:
  adapter: postgresql
  encoding: unicode
  database: f3_test
  pool: 5
  username: f3
  password:
Start the taps server on the sqlite database
$ taps server sqlite://db/development.sqlite3 user password
Migrate the data
$ taps pull postgres://f3@localhost/f3_development http://user:password@localhost:5000
Restart the Rails webserver
$ rails s
Cleanup the Gemfile
#gem 'sqlite3'
gem 'pg'
#gem 'taps'
$ bundle