BigV, I was involved with a project in Colorado that did just that. The digester produced a gas that was used to run two gensets: one used a modified diesel engine and the other used a Capstone microturbine. (I think they added a Stirling engine later.) There was a lot of sulfur in the gas, because of which the MT had to have lots of expensive gas cleanup equipment. The diesel engine just required frequent oil changes.

What is a capstone microturbine?

Read the article and see where they remove the hydrogen sulfide leaving 99% pure methane.

wrt Stirling Engines... I never managed to get my mind around how they work. But the diagrams are certainly elegant.

Now that I think about it, what role does the diesel play in the genset? Just a diesel engine that runs methane? I don't follow you there. Can you explain more?

Capstone is a manufacturer of microturbines. Here's their website: http://www.capstoneturbine.com/

The microturbine used in that installation was a 30-kW unit.

A genset is just an electric generator with something to spin it. This can be a steam turbine, a gas turbine, a gasoline engine, a diesel engine, a Stirling engine, etc. In this case, a Caterpillar engine was modified to operate on methane (~natural gas), then re-jetted to be able to use the digester gas, which was (I think) about 45% methane, with the remainder being mostly CO2 and CO. Natural gas is about 1000 Btu per cubic foot and digester gas is about 400 to 500 Btu/cubic foot, so the engine requires larger fuel lines or is de-rated for the lower-energy gas. (It's been about 5 years since I worked on this project, so some of the details might not be correct.)

In addition to spinning the generator to produce electricity, the engines also produce waste heat, which is routed back to the digester to maintain the temperature at around 105 deg F, so this is considered a cogen application.

You can read more about the AD project and see pictures here and here (links are to pdf documents).

Some of the larger dairy farms in VT are doing just that, with seed money from the state.
They use the methane to run generators for power and when the process is complete they recycle the bedding for another $50/$60 K savings per year.
It costs about half a million to set the system, up on a farm with sufficient livestock to make it practical.