Is it possible to lose fat and build muscle...

You could if your diet, supplements and training routine are 100%.

Other wise loosing fat requires a calorie deficit while muscle building requires a surplus (and progressive overload of muscle tissue via weight training programme).

That's why most body builders either bulk (gain weight and build muscle tissue - they bulk higher then required)

or Cut (loose weight\fat while trying to maintain muscle mass - over bulking helps here).

Absolutely.... very small deficits and Low carb!! Slow going but it can happen!

I would let Lyle answer this:

http://www.bodyrecomposition.com/fat-loss/adding-muscle-while-losing-fat-qa.html

^^agree, read the article.

In short, except in a few cases. the answer is no.