I was trying my first all grain brew yesterday when I realized that the beer was almost flat, which is weird because they spent about a week inside the bottle. It's a belgian golden ale and I've added about 10ml of a primming solution of 80g of sugar diluted in about 450ml of water (used an online calculator) for 33 cl bottles. What exactly has gone wrong? Was the yeast already dead at bottling or there wasn't enough sugar?
A week might not be long enough (especially if your yeast is particularly beleaguered, which would depend mostly on what the ABV of the finished beer is and how long it's been since fermentation).
Also, make sure your bottles are in a warm enough area (~70 °F is ideal for bottle-conditioning). Lower than this and it can definitely take several weeks, even with healthy yeast. I'd let it go a week or two longer and then if you don't see improvement, start to worry something went horribly wrong.
RE-editing: That might actually not be enough sugar. 10 ml of that solution would contain 1.5g. sugar (10g x [80g / 530g]). Using this calculator, at a beer temp. of 68 (a fair assumption), 1.5g sugar per 33cl (or 0.087 gal.) bottle would yield ~2.1-2.2 volumes of CO2, depending on what kind of sugar you use. Not super low, but also not super high either. I'd still wait and see if it turns around.
I don't think you have anything to worry about yet. Most of my batches take around 2 weeks to get to a reasonable level of carbonation, and are usually fully carbonated in about 3 weeks. Take Papazian's advice - Don't worry, have a Homebrew!
I would comment on the amount of sugar, but I can't tell the size of your batch from your post, but the standard for a 5 gallon batch would be 3/4 of a cup.
EDIT: My advice at this point would be to move them to a warmer room (at LEAST 65 °F), and wait another 2 weeks. If your beer is still entirely flat, there might possibly be something wrong with the yeast. My guess though is that it will be slightly carbonated, but not as much as you'd like. You could leave as is, or if you decide you want it more fully carbonated, you could just completely start the carbonation process over with more priming sugar (use roughly ~120 or 130g total this time).
Naturally, if there is no carbonation at all after 2 more weeks, you could move the beer back to a bottling bucket and re-pitch some more yeast along with the new priming sugar.
Good luck with whatever you choose. Let us know how it turns out!