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Not to worry... Corian drills easily with standard drill bits and hole saws and is moderately difficult to break. However you must be careful not to mar the surface when drilling.
Since these integral sinks have thickness that is not apparent from the top-view, don't assume that giving an inch clearance from the edge of the sink is enough... you may find that you can't install the mounting nuts because the underside of the sink is in the way!
Turn the countertop over and measure the shortest distance from the back of the integral sink to the rear of the countertop. Flip the countertop back and transfer this measurement to the front of the sink. If the countertop has a backsplash be sure to subtract from its thickness or you will locate the holes too close the sink. Locate the center of the sink holes 1" back from this mark.
(NOTE: If your countertop is not flat, be sure that the entire faucet is located on a flat area. Some of the "styled" countertops only give a little room for error.)
Now you should center the mark you made by measuring the width of the countertop and dividing by 2.
Depending on the size of the faucet, the center-to-center measurement for the holes may be 4 or more inches. Measure yours and transfer this dimension to your countertop. Most bathroom sinks require three holes... two for the supply lines and a center one for the pop-up control rod. The center hole should be exactly between the outer two. Double check your measurements by placing the faucet over the marks... you can never be too careful!
The drilling is done in two steps. First, drill a locator hole through the countertop at each of the three marks. This hole should be just slightly smaller than the size of the pilot bit in your hole saw. Using these holes as guides, use a sharp hole saw to enlarge the holes to 1 1/4 inches. I can't emphasize enough that you use a sharp hole saw. If you don't a few things can happen. The worst is that the hole saw will bind while drilling, skip out of the hole and destroy your entire day along with the countertop. A dull blade tends to generate lotsa heat, which can cause the plastic resin in the Corian to melt or burn. Furthermore excessive pressure needed to coax a dull drill could cause the Corian to break or crack. If you are not sure of the sharpness of that ol' hole saw you have laying in your workshop, go buy a new one.
Keep the drill speed moderately low... just apply enough power to keep the saw cutting at an even rate. If the saw seems to be bogging down reduce the downward pressure slightly and increase the speed a little. Don't exert too much downward force... let the sharp hole saw do the work. Remove the scrap Corian from the hole saw before making the next hole.
Be aware that drilling your own countertop could void your warranty. However, the cost of hiring a Corian-certified technician might be more than the cost of a replacement countertop!
Solid surface repair ("solid surface" is the generic name for solid resin composite countertops) is not an easy do-it-yourself project, primarily because repair materials are not readily available to consumers and there is a general lack of public information on repair techniques.
It is a shame, because skilled do-it-yourselfers should be able to do a great job on simple repairs. In fact, many woodworkers who have entered the solid surface installation business have found that solid surface machines very much like wood; many of the same tools and techniques are used.
There is a website that does offer repair materials. Art Specialties International at http://www.coriartspecialties.com sells Corian and Corian adhesives. Their market is to small fabricators of Corian bric-a-brac... but the adhesives used are the same as for countertops.
Both products on the page are used for Corian installations. I have not used either product, but I think that the Dupont Seam Sealer is the one you should try first. The "Cyberbond" is a cyanoacrylate... a close relative to super glue or "crazy" glue. Cyanoacryates are not designed to span gaps. Since a crack is a gap, you will not get much strength in your repair. The literature says it will work on two pieces that are clamped together. That is not the situation with most repairs.
The Dupont Seam Sealer is a two part epoxy, which will not only fill the crack but... if you are lucky with your color choice... will closely match the original countertop when done. This product is supposed to sand just like Corian, meaning that you can smooth it to the same finish you currently have by the use of fine sandpaper. Grades typically used on Corian are the following grits: 220, 300, 400, 600.
From my reading on the topic, a pro would use a random orbital sander with special disks designed for smoothing Corian.