Yes both can be done at once. Plenty of people have done it. You would probably need some Cardio mixed in with weight training I.E. 2-3 sessions Interval x 30 minutes and it wouldn't happen indefinitely.
You only need to read of people stating on the net that they are losing inches and lifting more to have real world testimonies and the people are usually those just doing some nromal blogging or posting on message boards, not people promoting some Myths, so when they fail to work for you, you get coersed into buying some supplements to correct your problem. Those people are called Bodybuilding magzine publishers
Don't forget that increasing thermogenic activity via weight lifting, could assist Fat burning and does increase Metabolic activity, but the Fat burning potential only lasts until your next meal, in which case the body would revert back to caloric intake.
The body would burn any calories from a previous meal, that it can use. It is not impossible to burn Fat and build muscle. One such thing Carnitine, which is created via the breakdown of Amino Acids, can help burn Fat to some extent, though supplemental Carnitine is reckoned to be not much good.
Plus if you repalced some of your Fat intake with less Fats that store easily like Saturates and replace them with Omegas and Coconut Oil MCT's, then you'd have a way to reduce Fat intake too.