Wikipedia says

The lowest-density aerogel is a silica nanofoam at 1 mg/cm3,[6] which is the evacuated version of the record-aerogel of 1.9 mg/cm3.[7] The density of air is 1.2 mg/cm3 (at 20 °C and 1 atm).[8] Only the recently manufactured metallic microlattices have a lower density at 0.9 mg/cm3.[9] By convention, the mass of air is excluded when the microlattice density is calculated. Allowing for the mass of the interstitial air, the true, unevacuated density of the microlattice is approximately 2.1 mg/cm3 (2.1 kg/m3).

So the evacuated version with 1mg should rise since air is 1.2mg.

Googling, I see one or two YouTube videos of aerogel floating in air. SEAgel apparently is sometimes used this way:

SEAgel is made of agar, a carbohydrate material that comes from kelp and red algae, and contains from one and a half to fifty milligrams of material per cubic centimeter of solid (in other words, it has a density of 1.5-50 mg/cm3). SEAgel can be made lighter than air using hydrogen - causing it to float or hang in the air.