Just replace the spout. Not too expensive and easy to do.
First you need to figure out how its attached. Look underneath. You are looking for a set screw or an allen screw. If you see one then loosen the screw and then pull off the old spout. Thats the newer style. If its an older spout you may have one that screws on. If no set screw then just twist it off - Lefty Loosey, Righty Tighty.
Take the old one with you to the hardware store and buy a replacement to match.
Sounds too simple, right? Well - If it IS the kind screwed on it could be on there so tight that you think you are going to bust something getting it off. Or, you could not see the set screw, think its a twist on and try to twist off the set screw style. That could mar up the pipe and cause a new one not to seal properly.
My guess - you have a set screw. Inside the spout is an 'O' ring. That O ring is whats leaking. If the spout comes off easily and seems well made you may want to see if you can find a new O ring and keep the spout. The replacement spouts at the home center are cheap plastic junk. O ring should only cost a nickle but the home center will sell them in an assortment for $4.
Good luck - post back if this is the correct fix.