Actually this shouldn't be too hard, provided you have an accurate scale. You can simply dissolve some white table sugar in your mead and compare the reading you get with the reading you would expect to get if you had dissolved the same amount of sugar in water instead. The difference will tell you how far below SG 1.000 (density of water) you are.
Say you took 5 grams of sugar (sucrose) and dissolved it in 95 grams of water. This would give you a 5% solution (5 grams of sugar out of the total weight of 100 grams). This is the same as 5 degrees Plato or SG 1.020 (for all intents and purposes, degrees Plato times four gives you the last two digits of SG [5 x 4 = 20, hence 1.020]).
If you used your mead instead of water you should measure a lower number. Say you added 5 grams of sugar to 95 grams of the mead, took a gravity reading, and got 1.008. Since you know using water should have given you 1.020, you can assume you started with liquid that was 0.012 SG lower than water (1.020 - 1.008), or 0.988.
It may be more trouble than it's worth, but it should let you know the gravity to a fair degree of accuracy. It would be best to do everything at room temperature, both for accuracy of hydrometer measurements and to avoid evaporating any liquid.
I should have noted that degrees Plato is the same as the percentage of sucrose in aqueous solution (i.e. 12.5 degrees Plato is the same as a 12.5% sucrose solution). Obviously mead/beer/cider/whatever have different sugars in solution, as well as alcohol and other soluble materials, but they are measured as the equivalent weight of sucrose.
The formula to add enough sugar to a given amount of liquid to make a 5% (or any percentage) solution is as follows:
Ws = [ % * Wl ] \ [ 1 - %]
Ws = weight of sugar to add
% = desired percentage of soultion (expressed as % / 100, so 5% would be 0.05)
Wl = weight of liquid to which you will add the sugar
And working through a hypothetical example: you fill your hydrometer tube enough for a measurement and the weight of that liquid is 162 grams. You want to add sugar to make a 5% solution, which should get the hydrometer to read in range. So the weight of sugar you want to add is
= [ 0.05 * 162 g ] \ [ 1 - .05 ]
= 8.1 \ 0.95
= ~8.53 grams
And let's just check our math quickly: our resulting solution weighs 162 g + 8.5 g = 170.5 g.
8.5 / 170.5 = 0.0498
or basically 5%. Cool!