Your Sinatra app:

get '/' do
    callback = params.delete('callback') # jsonp
    json = {'your' => 'data'}.to_json

    if callback
      content_type :js
      response = "#{callback}(#{json})" 
    else
      content_type :json
      response = json
    end
    response
  end

Your HTML:

<script type="text/javascript">
    function parseResponse(json) {
    // Do something with the data
    }
    </script>
    <script type="text/javascript" src="http://xxx.com/?callback=parseResponse"></script>

You can also do the same with jQuery:

$.ajax({
    type: 'get',
    url: '/',
    dataType: 'jsonp',
    success: function(data) {
      parseResponse(data);
    }
})