Weigh is an innate command that gives you a rough estimate of the weight of an item. According to the help file, its accuracy is based on dexterity. If a more accurate weight is needed, it is recommended to use a balance or other weighing device such as the scales at the Djelibeybi post office.
You can generally only weigh one or more items if you are carrying or holding them, but not while wearing them. Weighing multiple items will give the individual weights then total weight.
In a few rooms you can use items to get a more accurate weight.
Heavier items cannot be weighed on the balances and so can only be weighed on the scales with a little less precision.
You can weigh many items at once on a balance or scales, but not worn items. The balance will give the individual weights then the total weight and scales will only give the individual weights.
weigh <item(s)> on <balance/scales>
- Balance in pound (lb)
A balance is a device used for measuring the weight of items with pound (lb) as the unit of mass.
- Balances giving the weight in lb are available in the Ankh-Morpork post office and in the Mayor's office upstairs in the Escrow town hall.
- To use the balances, you need to have completed a quest at one of the balances.
- Items heavier than all the weights (above 13 4/9 lb) cannot be weighed on the balance. You can try to have them weighted on the djelian scales.
- Items with weights not perfectly divisible by 1/9 lb are expressed as a range (e.g. "less than 1/9 lb" or "between 8/9 lb and 1 lb")
- By weighing multiple items having a total weight that is a multiple of 1/9 lb the exact weight can be determined.
- Note that Discworld uses a slightly lighter pound, equal to exactly 450 grams. 1/9 lb is thus 50 grams.
- The maximum precision is 1/100 of a gram (0.01 gram).
- Jeweller's balance in gems (g)
This balance allows you to weigh very light objects with gems (g) as the unit of mass, which coincidentally gives the same abbreviated unit as the roundworld gram.
- There is such a balance in Schmit and Weestone's forge on the second floor above the ground floor of the Academy of Artificers in Sto Lat, just south of the gymnasium and Mr Ritservals (fly-to), which is outside when the dome is retracted (pull lever). 
- 50 g = 1/9 lb ; 450 g = 1 lb ; 1 g = 1/450 lb
- Items heavier than all the weights (above 200 g or 4/9 lb) cannot be weighed on the balance. You can try to have them weighted on one of the lb balances or the djelian scales.
- Items with weights not perfectly divisible by 1 g are expressed as a range (e.g. "less than 1 g" or "between 1 g and 2 g")
- By weighing multiple items having a total weight that is a multiple of 1 g the exact weight can be determined.
- The maximum precision is 1/100 g (0.01 g).
- Pair of weighing scales in deben
A pair of scales that give the weight in deben are available in the Djelibeybi post office north of the bazaar.
- Each weighing attempt costs 1 talon.
- Using the scales accurately requires people.culture.djelian, a 150 bonus seems enough to always succeed.
- 9 deben = 2 lb ; 1 deben = 2/9 lb ; 1 lb = 9/2 deben
- Due to the conversion rate, the weights returned by the scales introduce a 1/9 lb margin of error.
- It costs one talent to weigh something on the scales, you need to have the appropriate coin, no change is made.
Description of balances
Balance in pound (lb)
A weighing balance. This is a largish bronze balance, securely bolted in place. The main part of the balance is a long arm which pivots at its centre. There is a pan hanging from each end of the arm such that it will be level when the weights in the pans are equal. A third pan is fixed to a stationary part of the balance where the weights can be held when not in use. The middle pan holds a small weight, a large weight, a very small weight, a very large weight and a medium weight. The left pan is empty. The right pan is empty. The left pan hangs level with the right pan. You could probably use the balance to weigh something. There appears to be something written on it. You read the weighing balance: Written in Uberwaldean: This balance accurate to one ninth of a pound.
Jeweller's balance in gems (g)
A neat little jeweller's balance. Small and delicate, this jeweller's balance looks like it wouldn't be able to weigh anything too heavy. A stack of tiny precise weights, marked in "gems", sits next to it. From the markings, a "gem" seems to weigh 1/450 of a pound. It appears to have something written on it. You read the neat little jeweller's balance: Written in neatly printed letters: 450 gems (g) is a pound (lb). Maximum weight: 4/9 pounds (200 g).
Pair of weighing scales in deben
A pair of weighing scales. The scales consists of two pans supported from below, on top of a closed wooden box. A basket is also secured to the box, and occupied by carved stone animals of different sizes (and probably masses, too). A small pictogram hints how this contraption could be used to weigh things. It looks like the scales will munch talon coins for each item weighed. It appears to have something written on it. You read the pair of weighing scales: Written in neat hieroglyphs in Djelian: For the convenience of ignorant foreigners, it is known under the bubble of the Sky Goddess Nesh that two "pounds" of sand weigh nine deben.