I'm more familiar with the use of a pycnometer for the volume of liquids, and most of the gas ones seem geared towards powders/porous materials, but calling some of the manufacturers should point you to an appropriate instrument.
Failing that, simply follow Archimedes' idea: place the sample in a full beaker of (degassed, distilled) water, collect the overflow and weigh it- since you know the density of water, you then know the displaced volume. I bet you could get 1% accuracy this way, easily.
Perhaps you could actually manufacture a cuboid to high dimensional tolerance (for volume) and then measure its mass. Using a 'Eureka Can' isn't a very accurate method of volume measurement.
There may be other approaches, like measurement of the speed of sound in the material. You would still have the problem of measuring dimensions unless you could somehow measure the reflection at the interface with a known substance.