Is it normal for a 12 week old weimaraner puppy to constantly bite?

The fact you got the dog at only 4 weeks old is worrisome. I smell a BYB. You should enroll your puppy into puppy kindergarden. Doing a puppy kindergarden class should give you the proper knowledge on how to deal with your puppy's biting issue and gives your puppy VITAL socialization skills. And YES it is common.

After puppy kindergarden be sure to get some oobedience training to help with any lingering issues. Your dog may have MANY issues if you truely got the puppy at only 4 weeks old. The bond you share is going to become more important as time wears on.

Weimaraners are active dogs and are not suggested for an urban lifestyle. Finding your weimaraner an outlet for it's energy may prove important.
References :

4 WKS!?!?! Sheesh, no wonder you're having behavior problems. You are NEVER supposed to get a puppy before it is 8 wks. The longer the pup stays with its mother and litter the better for its development. Your dog has not learned the things it would have from its litter mates and mother since it was taken so early. The pup lacks proper bite inhibition. You will have to teach the pup not to bite or mouth people.
Every time the pup jumps or mouths you, yelp real loud, and then walk away while ignoring the pup. If it follows you, go somewhere you can watch it, but it can't get to you. This will teach the pup that putting its mouth on someone makes the fun go away.
I also recommend teaching your dog to sit for attention. Do not pet the pup unless it sits. If it jumps, simply ignore the pup until it sits.
References :

Part of your problem, is that the dog left her litter a month too soon. She didn't learn about bite inhibition (as she would have, playing with littermates) nor was she socialized with humans & her breeders who should have started working on discouraging biting between 4 and 8 weeks of age.

You need to work on TEACHING bite inhibition. I can provide links on this. If you don't get the biting under control & the JUMPING (both are dominance issues) you WILL have permanent problems. She really isn't teething until 16 weeks, when new teeth start coming in. She doesn't understand NOT to bite, or about biting less hard (which comes first.) HOW you interact & play with her, can help or hurt the cause. Don't play tug of war with a biting puppy, don't wrestle or play roughly, period.

To help you, I suggest you go buy Bitter Apple or Fooey in a spray at your local pet store & apply to your legs or any part of your body or clothing she is biting on. It won't stain, but tastes AWFUL to most pups & therefore will help discourage her, while you also tell her VERBALLY…"NO BITE!" at the same time she is trying to bite you. The bad taste wil help her learn not to bite you, but you also have to work on it & re-enforce the "NO BITE!" reminder, if she tries to bite other people. Collar correct & make her stop, at the time you make a verbal correction. Say it like you MEAN it! This biting behavior can get you sued, if she hurts somebody, as her jaw strength increases.

Do enroll her (with you) in puppy or beginner, obedience classes. You've got a dominant breed & this puppy may need more training than just one class, but you need to do the work, if you want her to mind and respect you. (This is part of your responsibility as an owner.) She needs the socialization with other puppies, & she needs to learn basic manners or she will become a juvenile delinquent. BTW, 90% of dogs surrendered to shelters, never got any training.
References :
25 years in Weims, multiple titles on every one, I've ever owned. (Well behaved dogs can go anywhere.)
Links on bite inhibition:
http://www.canismajor.com/dog/bite3.html
http://www.canismajor.com/dog/meanit.html
http://www.doglogic.com/apdtfaqs.htm#Bite (see articles by Walton & Richards)
http://www.diamondsintheruff.com/biteinhibition.html
http://www.canismajor.com/dog/bite2.html

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