We have a client with the unfortunate situation of losing access to their current domain name "A".
We've just finished finished setting their website up on domain "B".
Both websites on "A" and "B" are the same.....
However, "A" is no longer available and we don't have access to renew it. It's now serving advertisement!
That last word "same" is where things get interesting. Because neither copy of the site exclaimed that its a duplicate to google via the canonical value in the HTML link tag, google will at best be confused as to which domain has the original copy, and because of this, rankings will suffer at best. I'm not sure how long both copies of the site were active, but in that time frame, google would be unhappy for seeing copies, and now since site "A" is all advertising, the rankings have basically plummeted, but the good news is that there's no issue with duplicate content.
More specifically, how can we tell Google that the old domain is lost...
In terms of your website topic, the old domain is technically lost to google because google sees it as a bunch of advertisements.
....and here's where you can find the new one. ... How can we preserve SEO juice and rankings from the old domain to the new one?
If it is possible, copy everything over from site A to site B, and have site B up and running the way site A was before, then access Google Webmaster Tools and verify the new domain and add a sitemap for the site, making sure all URLs in it belong to the new domain, and not the old one. Then select the gear icon in GWT and increase the crawl rate. Google will be able to pick your site up within a few days.
Just make sure if you have access to any section of site A after copying files that you make it inactive to the public, or better yet, add redirects from site A to site B. If you don't take any action at all on site A, then google will flag both site A and site B as having duplicate content if site A ends up being active again (instead of serving advertisements).