chazza is correct almost all phone do offer this feature now, however beware that the sim only stores name and one phone number per entry, so any advanced contact details (i.e. multiple phone numbers per entry, address, email address, birthday details etc etc) that you may have in your current phone will be lost.
if your new phone is same make as old one then you may be able to back phone book up onto a computer and then restore it to new phone, nokia pc suite and sony erricson pc suite both will do this for sure, not sure on other brands.