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Exactly as per the title. This is surprisingly hard for me to google.
I found one estimate of about 10 pounds, and the tech at my LBS "guesses" that a standard 30 x 8 x 53 box weighs 2 pounds. For long distance shipping, that much of weight difference can save a lot.
First, I'll give you an estimate on the weight of a cardboard box. Then, you can read the side notes at the end of this answer to see why the weight is a relatively irrelevant quantity.
EDIT: This link sells a bike box and lists the weight as 7.40 lbs. The rest of this answer gives you a way to estimate this, as well as tells you why this whole problem is irrelevant from a cost perspective.
According to this link, the density of cardboard is approximately 30-90 kg/m^3 or 0.03-0.09 grams per cubic centimeter.
Now, lets estimate the amount of cardboard in a box. Say each side of the box has thickness T centimeters, and the sides of the box are L,W,H in centimeters. Then, the volume of the cardboard in the box is 2*T*(L*W+L*H+W*H), so the mass of cardboard in the box is density*volume of cardboard or between 0.03*2*T*(L*W+L*H+W*H) and 0.09*2*T*(L*W+L*H+W*H) grams.
Now, lets estimate the dimensions of a box to ship a bike. According to this thread, the typical bike box is about L=130 cm long, W=18 cm wide and H=79 cm height. For thickness, to make the numbers a bit easy, lets say the cardboard is T=1 cm thick (this is significantly thicker than most bike boxes I've seen).
Plugging into the formula, we get a range of approximately 800-2500 grams for the box (1.75 to 5.5 pounds).
I suspect the thickness of the box is closer to half a centimeter (which would halve the estimate) and the density is probably on the higher end of the scale (but probably not 0.09 grams/cm^3). So, I'd guess the weight is closer to 3 pounds.
The simple solution is to take a scale, and a bike box and weigh it.
Side note 1: The dimensions from that thread are only slightly different from your box, and the values come out to be nearly the same.
Side note 2: With carriers like UPS and Fedex, the expense of a shipping bicycle is not primarily the weight, but the fact that a bike box has large dimensions, so it can count as an oversized package [and paying the LBS to pack it]. For small packages of the same weight, note that the cost of shipping will be significantly lower (go to the UPS shipping calculator and put in your source and destination with a small box, say, 10 in x 10 in x 10 in weighing 50 pounds versus a bike box weighing 50 pounds). UPS calls this dimensional weight, so for your bike box, it will charge you assuming that your package weighs ~75 pounds even if your bike weighs 10 pounds packed. So, I guess you can fill your bike box with some rocks or something cause you're paying for 75 pounds even if you don't use it anyway!
A bike box measuring 51.5"x7.5”x32" weighs 7.3 pounds
As weighed on digital platform scale with accuracy to 0.05 pounds
No mystery here; I simply weighed a Specialized bike box (Allez Sprint came in it). My workplace shares a building (and recycling bin) with a bike shop. It was much easier and simpler to weigh the box than for me to do a bunch of calculations or google searching. And the answer is really about the same as Batman's answer above (before all the calculations.). My box weighed 7.3 pounds. For estimated BikeFlights shipping weight purposes I added the weight of my bike as shipped to the weight of the box and added a couple pounds for packing material.
