European disposable cell phone

Mary, I'm assuming you're referring to an inexpensive prepaid Cellphone? Are you planning to "dispose" of it by throwing it in the trash when you get home? If you buy a phone in Denmark, you will be able to use it in Switzerland and Italy, although the rates will be higher since you'll then be "roaming". Therefore you'll deplete your "bucket of minutes" more quickly. Roaming rates are capped by the E.U. but the costs will still be higher outside the country where the phone was purchased. You could also have some issues topping-up when you leave Denmark. You might also have a look at the "travel phone" firms such as Roam Simple, Cellular Abroad, Call In Europe, Telestial, Mobal or EuroBuzz. That would allow you to have a working phone as soon as you step off the plane, consistent rates throughout western Europe and post-paid billing so no concerns with topping-up (features and rates will depend on which of those services you choose). Happy travels!

My cell phone carrier suggested I purchase a SIM card for my phone and replacing the current SIM card when I arrive n Europe. Your phone will have to be unlocked to do this. Now sure how all this would work.

Janie, it works pretty much as per your description. Get a SIM card that works in Europe - either one from a travel phone company or one from a local European company - and you'll have a working phone in Europe. This will be totally separate from the phone account you use at home. You'll have a new phone number, new voicemail, etc. Calls, text messages, and data will be governed by different rates from whichever company you acquired the SIM. If you have any specific questions about the process, I know there are many here who would be happy to help explain things further.

@Janie, In order to provide more detailed suggestions, it would help to have some further information. For example, which cell network are you with, which model phone are you using, is your phone unlocked, etc.