When I initially wrote this post it was possible to do everything in one day, but now, because of changes to the express service at the MFA, the process will take a minimum of 2 days. If you do not use the express service, you should plan for 4-5 days to get the entire marriage process completed.
The Ministry of Foreign Affairs (MFA) offers the express service if you submit your documents before am. There is a new rule that requires you to have at least 10 days left on your 30-day entry stamp, or tourist Visa.
Alternatively, get in a taxi and go straight to Chaeng Wattana to the Department of Consular Affairs.
If you do not have 10 days left of permission to remain in the country, then you need to go to immigration and get a Visa extension before you can start the marriage process.
Don't dilly dally: choose a translation centre quickly and make it clear you are on a deadline. They should be able to do the translations within 1 hour.
If they say they can't do it so quickly, let them know you can pay a little extra.
This certificate, if not in English, must be accompanied by an English translation.
Only the original evidence or an official certified copy will be accepted – photocopies will not be accepted under any circumstances. If your embassy offers a walk-in service, get there at opening time.