It is important to have a good running base with appropriate training and experience over a period of time before entering the half. If you just started running a little over a month ago, then I would discourage you from jumping directly into major distance competitions. Take a look at some the level of commitment/advancement required for some of the typical schedules to prepare for the half, and I think you will find that the demands are too extensive to pursue right away.
I would say see how your 10k goes and then look at other race possibilities (5k, 8k, 10k, 15k range) and then also devote the time to finding the best training options for you. Everyone is different, but getting to the half within a year is a good accomplishment.
If you just started running 6 weeks ago, that gives you 3.5 months to train. It is not unrealistic to run a Half Marathon IF you find a good trainng program for beginners (most are 10-12 weeks) and strictly abide by it--which can be tough for first-timers. It also depends on what type of Half Marathon you intend to sign up for, because some are certainly more challenging than others due to terrain, climate, etc.
If you're just looking to finish a HM without worrying about your time, then it is doable as long as you train hard and do your research. I'd go with Jasko's suggestion by seeing how you fair in the 10k before making the committment, but don't let that deter you from training now.
If you're willing to do a walk/run program, you could be ready to complete the race. If you want to run the whole thing, then you really risk injury by trying to do too much too soon., though it may be possible. If you want to actually race it, then it's too soon. From everything I've read, a HM is best done when you have a base mileage of 30-50 miles a week. Getting to that base mileage takes time, especially if you want to avoid injury. The usual recommendation is to only increase your mileage at 10% a week.
I'm an example of how not to do it. Seven months after I began c25k, I was supposed to do a HM. A week before the race, I developed a pelvic stress fracture. Five months of no running, and I'm out the cost of two races that I wasn't able to run.
People will hate me but I say "go for it!" It will probably increase your mileage more than the 10% of the suggested weekly amount but it will also do two things. 1) get you to push yourself harder than you ever have gone and 2) provide a safe, controlled environment for you to run that distance.
I have a hard time with my long mileage training runs because there is NO support for me. No water stations, no blocked intersections, no medical staff on hand. I am out running 10+ miles on my own and should something bad happen, well then "oh well". You'll know if you are over doing it and will slow down If you feel good, keep going! You will be so very happy with yourself in the end! (No doubt super sore the next day too!) :-)
Let us know how it goes!