I received this question from one of you. The choice is yours and it depends on your personal goals: both long-term and goals for the given day!
Do first the part which is more important to you on a given day or as a part of a given plan. If you aim at building muscles, lean body mass, power and strength, then warm up thoroughly and do the weight training session first. This way, your muscles will be still kicking and alive for the weight lifting part, while - in case you do the cardio first - your strength and overall muscle capabilities would be temporarily lowered due to fatigue. If you would rather put accent on your cardio, endurance, or, for example, improve the speed, then warm up and do the cardio first. Only then, top it up with some weight training. This sequence, actually, seems to be burning more calories than the other way around.
I am always for the variety and for changing and shaking things around in terms of the training. So, if you are pursuing general fitness, I'd simply recommend that you do cardio first on some days and the weights first on the other days. Only this change would assure that your body will keep guessing each and every day, what is next. Last but not least, I would not give up the weight training part. This is what builds the healthy muscle tissue in the body, which, in turn, burns so much more calories per hour than any other body tissue (such as fat). Once you get some nice muscles and train them, following each training session, you've got a guaranteed rest of the day of increased calorie burn (even while doing not related to fitness activities). The best is to perform your training (both cardio and weights!) using the interval method, which also guarantees you an increased post-exercise burn for up to 36 hours after you visited the gym! Stay Fit-Fit and keep me posted at email@example.com.