I bought some kitchen scales on eBay, they weight down to the gram and I just sit Squawks in it to weigh her. Vets often wrap them gently in a towel first if they don't want to stay in the scales, but you have to weigh the weight of the towel first so you don't include it.
I did the same as Siaarn- I bought a kitchen scale. I usually have to put a cheerio or some treat on the scale to get Sunny to stand on it long enough for me to see his weight. They do sell scales with perches attached but I think they're pretty expensive compared to a regular kitchen gram scale. (I paid $15 for mine.)
My tiel and green cheek will stand on the scales themselves... sometimes my 'let will too but not always... to weigh the 'let and the budgies I place a small travel cage on the scale and tare it out... then pop in a bird and the weight displayed is the weight of the bird... my small birds are all unafraid of hands, but don't like to be grabbed, but if I go slow and steady I can reach in the cage and pick up the bird I want without upsetting them to much.
I use digital kitchen scales. They have a nice plastic container that fits on the top that isn't too high and it nice and wide and I sit our birdie in there. The thing is to get them to sit still long enough to weigh them lol.