I don't know how effective it would be for weight loss, but you sure would be hungry and miserable! The great thing about calorie counting, imho, is that it gives you plenty of flexibility along with a very strong incentive to eat healthily. Two snickers before lunch for someone on a 1200 calorie diet will leave them starving by the end of the afternoon with no calories to spare. On the other hand, a lovely big egg white omelet packed with veggies will make you happy and full and healthy without breaking the calorie bank.
I know you didn't ask for suggestions on this, so you're welcome to ignore me completely. But you might want to consider the possibility of starting with a higher calorie allotment. I bet you could easily lose on far more than 1200 calories a day. And really, why restrict calories unnecessarily?

Here's the math: at your weight and height, depending upon your age, you can maintain your weight somewhere on between 2000 and 2300 calories a day, even if you spend all day sitting at a desk. One pound of fat is 3500 calories, so to lose one pound in a week, you'd eat 500 calories less than maintenance a day (so, 1500-1800 calories a day). If you exercise, this will increase the calorie deficit, so you'd probably lose between 1.5 and 2lb a week on a 1500-1800 calorie diet with regular cardio. Those extra 300-600 calories could be used on fiber and protein, both of which will help you to feel full and happy, contribute to your general health and increase your chances of staying on your diet.
The other reason not to start out with a 1200 calorie diet is that it leaves you nowhere to go when you inevitably reach a plateau. If you eat 1600 calories a day for a couple of months and then plateau with 10lb still to lose, then you can drop your cals down to 1200 or 1300. If you're already eating only 1200, you can't drop them any further and it will be frustrating.
Just a thought