I am by no means an expert but why not just sharpen it yourself? Whatever you do, please don't use a bench grinder and read through this:
http://www.fs.fed.us/t-d/pubs/pdfpubs/pdf99232823/pdf99232823Pdpi300.pdf
Page 27 starts to go over sharpening but just read the whole thing. There is a template for a decent bit (blade) profile (p. 29) and lots of other good stuff in there. Proper sharpening means filing or moving the stone TOWARDS the blade, not away from it --and they say a good, sharp axe can shave the hair from your arm. Wear proper footwear when you chop/split and take a good look at my left thumb in the two photos below. Unless you like to bleed, wear gloves no matter how dull you think the blade is. It'll get sharp quickly and all it takes is one slip and you'll be bleeding.
Post up some photos of the axe. Are there any logos, markings or engravings on it? I've not ever heard a bad word about GB, as expensive as they are people seem to say they are worth every penny. That said, there are a lot of very inexpensive, vintage axes out there that can be just as good as a GB. Flea markets, yard sales and eBay are good places to look.
Convince your dad to take you back to the hardware store you bought the handle at and pick up a few more things. First, pick out a pair of leather work gloves, then get a quart of boiled linseed oil (BLO), a few of those cheap chip brushes (throw away paint brushes), a few files (get at least a bastard and second cut, see link below), A PROPER FILE BRUSH, a sharpening stone, and some sandpaper (couple sheets each of 80, 100, 120 & 200 or 220 grit). Your dad may already have some of this stuff (files, sharpening stone, sandpaper, chip brushes).
https://en.wikipedia.org/wiki/File_(tool)#/media/File:FilesFlat-Smooth-2ndCut-Bastard.jpg
You can get all of that for probably around $50-$60. Take a look at the bit profiles in "an axe to grind" so you have a good idea of what profile you want on your axe. Put the axe securely on a workbench and then start with the bastard file. With the bastard file you really just want to remove the rust, burrs and dings so the 2nd cut file will work well. Use the second cut file to shape and profile the bit, and then either run over it quickly with a fine cut file or switch over to a course (and then fine) whetstone and have at it. It may take a couple hours but go slow, be patient, don't press too hard with the files or the stone and take pictures and post them up. You'll get some really good feedback on your progress.
After you've sharpened the axe, wipe it down with a rag to get all of the rust/metal shavings from it and then have at it with the sandpaper. I assume the handle you bought came with lacquer on it. Sand off all of the lacquer/varnish, starting with either 80 or 100 grit, then move to 120 and 200/220. When you are down to bare wood, apply BLO. One of those rectangle, plastic containers from Chinese takeout works very well for this; pour a little bit of BLO in the bottom of the plastic container and put the axe head in the BLO. Take a throw-away paint brush and start brushing the BLO all over the handle, especially on the end/knob. You'll see how fast the wood soaks up the BLO, especially at the end into the grain. Keep going back a couple times a day and painting the BLO onto the wood. Don't worry about the axe head, the BLO will dry and then wear off with use. Keep giving it BLO until it stops taking it. It may take several days until it stops soaking it in and depending on the temperature, it may take a few days after that for the BLO on/in the handle to fully dry but it'll be worth it.
**Linseed oil dries through a process of oxidation (like rust). When it oxidizes, it generates heat and can cause a fire. Be careful not to throw piles of oily rags (or even paper towels) into a trash can without laying them out and letting them dry first, or putting them into a fire-safe can. A bucket of BLO exposed to the air isn't an issue, many folks keep their axes stored head-first in a bucket of BLO to keep the top of the handle (eye) from drying out. Just don't throw a bunch of rags soaked in BLO into the trash, this is why I suggested using a brush rather than rags to apply the BLO as you can keep the brush in the BLO and use it over and over.
Once the axe is sharp, please do carry and use it correctly. A sharp blade is a safe blade but any blade can be dangerous. Last thought, if the axe was your grandfather's, why not ask your dad if he knows anything about this stuff?
Hopefully someone corrects me if I said something wrong/inaccurate.
