The cost depends on the embassy, but will be around the 3,500 Baht mark. If your embassy has the affirmation form online, print it out from embassy website before you go, and fill in the relevant sections.
For the British embassy, you will need to print the form.
In other words, simply having a ceremony in Thailand doesn't mean you are legally married.
It is very possible to complete the process of getting the correct documentation together in one day, and that's the goal of this post.
If you walk to the top of Wireless Road (back toward Pleon Chit BTS station) and up over the bridge onto the other side of the road, you'll be standing outside a language school called Language Express. Along this side road you'll find a number of reputable translation services.