Tracy, when babies are born prematurely, their development tends to follow the age they would be if they were born on their due date. For your 2-month-old baby who was born 1 month early, you can expect her development to be like that of a 1-month-old. Remember, though, each baby is unique and there's a wide range of what's normal for babies' development at every age. Also, if your baby had any medical problems at birth, there might be different expectations for your baby's development.
What might you observe in your baby between 1 and 2 months of age? She will need to feed frequently, typically every 2 to 4 hours, and probably still needs to wake up at least once at night to feed. Although she has probably spent a lot of time sleeping during the day and night, she will start to spend more time awake and alert during the day. Your baby's vision is best within 8 to 12 inches, so she can see your face best when it's close up, like when you feed her. You might even notice that she follows your voice, your face and baby rattles with her eyes. Don't worry if she appears cross-eyed—it usually takes a few months for the eyes to focus together.
Your baby is also likely to start smiling during this time, which is wonderful. You might also notice that when you talk with her, she coos, gurgles and moves her arms and legs to communicate with you. As a newborn, her head and neck were very floppy and you needed to constantly support them, but now you'll start seeing that she has more control of her head. She might hold her hands in tight fists and make jerky movements with her arms and legs, but you'll start observing that her movements become smoother, her hands will begin to loosen up, and she will start to bring her hands to her mouth or in front of her eyes.
Enjoy this precious time with your baby! Also, don't be alarmed if your baby has longer periods of fussiness or crying, especially in the evening. This is completely normal at this age. If you have any concerns about your baby's development, talk with your doctor about it.
Our parenting advice is given as suggestions only. We recommend you also consult your healthcare provider, and urge you to contact them immediately if your question is urgent or about a medical condition.