When I comprehend it, the Sinatra framework, unlike Rails, doesn't offer an ORM. For the reason that situation, how can you speak with a DB inside a Sinatra application? Or perhaps is Sinatra just for applications that do not make use of a DB?

If you are using Sinatra, I can not recommend DataMapper highly enough. I've got a couple Rails applications where I am tied to ActiveRecord, and I am constantly cursing its weak points and design defects. If you are on Sinatra, DataMapper is an extremely viable option.

require "rubygems"
require "sinatra"
require "datamapper"

DataMapper.setup(:default, "sqlite3::memory:")

class Post
  include DataMapper::Resource

  property :id,    Integer, :serial => true
  property :title, String
end

Post.auto_migrate!
first_post = Post.new
first_post.title = "First!"
first_post.save

get "/" do
  Post.get(1).title
end

If you want ActiveRecord, use that. Or something like that else. Datamapper, for example. For AR with SQLite, this works:

require 'rubygems' # may not be needed, depending on platform
require 'sinatra'
require 'activerecord'

class Article < ActiveRecord::Base
end

get '/' do
  Article.establish_connection(
    :adapter => "sqlite3",
    :database => "hw.db"
  )
  Article.first.title
end