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I think the only buses that go from Dubrovnik to Montenegro are departing from Dubrovnik Center, which means that you need to go back north from the airport to reach the center, whereas Montenegro is south and pretty close to Cilipi airport in fact, so it is probably less of a hassle (price and time) to use private transfer services to reach Montenegro.

Prices seem to vary from 80 to 120 EUR.

It could be a good idea to reserve the transfer in advance in July and August considering that many visitors to Montenegro arrive in Dubrovnik airport.

Dubrovnik airport is not well served except for the local bus into Dubrovnik.  I have heard of people waiting out on the road to stop the Montenegro buses coming through from DBV but can be risky.

I have taken a Globtour bus from Montenegro to Dubrovnik in 2010.  It leaves daily at 1500 from Kotor.  The driver allowed us to get off at the airport (however, we later learnt that someone getting off at the airport took the wrong bag, so they may not be willing to do that again).    I can't locate the route on Globtour's website but by following a link from Globtour, I could book it at adriatouring.com/BookingStep1.aspx

You might also like to look at libertasdubrovnik.com/ and follow the link to the right 'Sorted by company'   (libertasdubrovnik.com/voznired2.pdf) and you will see bus services between Dubrovnik and a number of points in Montenegro.  Herceg Novi is the nearest point to the border crossing and from there you can get buses to various points in Montenegro.  You can follow up some of these companies on the web, e.g. Blue Line (blueline-mne.com/internationaltransportation.html)

If there are a few of you, taking a taxi is probably the easiest.

Enjoy yourself.  The hassle will be well worth it!

As far as I know, Croatian (official) taxis are not allowed to cross the border and vice versa. This is why you might want to book from a private transfer company.
Have fun!

gearodin wrote:

If there are a few of you, taking a taxi is probably the easiest.

Enjoy yourself.  The hassle will be well worth it!

Yes, I should have been more specific. It should be a preordered pickup service!  Sorry about that.

You have one travel agency from herceg novi (montenegro) that organize transfers from dubrovnik airport to montenegro. there web site is intravelmontenegro.net, the prices are ok, and the services is nice

I heard there is now a bus departing from Tivat bus station at 8 in the morning goes up to Dubrovnik stays for about three hours then comes back again to Tivat.