The short answer is yes, you can definitely gain muscle and lose fat at the same time. This is particularly true for new lifters, who are often able to build muscle while eating at a calorie deficit.
But a lot of it will depend upon your diet and the kind of lifting that you do. What kind of calorie intake are you currently working with? Your weight is pretty good right now--I'd bet that if you ate close to maintenance (or perhaps just slightly under--but clean, high protein, etc) you'd be able to re-comp your body in the way that you want.
The only thing is that you will probably need to use a different metric than the scale to gauge your progress. Take measurements and/or go to the gym and have your body fat tested by an experienced tester. If you're gaining muscle, that will show as a gain on the scale (ie, muscle is 'weight').
You might want to check out a program like Female Body Breakthrough, which is geared towards an audience that is normal-weighted and wanting to lose fat and build muscle. New Rules is also excellent. Either way, include HIIT and metabolic elements (like kettlebell swings) in your workouts. Those help you to burn fat.
(And come over to the Weight Lifting sub-forum--folks there will have opinions and advice for you!)
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