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There are many options, depending on where you live and where you'd like to go on the way to Patagonia or on the way back home. At present, internal flights in Argentina are expensive and some have lower baggage allowances for the domestic sector of your journey. The first low-cost airline has just been granted a licence to make internal flights in Argentina and others are waiting in the wings, but they will make their first routes to the most popular tourism destinations, including Cordoba, Mendoza, El Calafate and Iguazu, and are unlikely in the short-term to offer low-cost fares to Welsh Patagonia.
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Non-stop flights are available from the UK via British Airways and Norwegian, but neither operates internal flights in Argentina yet, although Norwegian is planning to. Most European airlines have a one-stop offering to Buenos Aires. LATAM, the combined airline of LAN Chile and TAM of Brazil has the most flexible offerings. For example, at certain times of the year, they have flights from Bariloche to London with a single stop in San Paolo. Buying a return ticket from London to Bariloche (300 km north of Esquel and a great place from which to start your Patagonia adventure) is normally the same price as a London to Buenos Aires return flight, meaning that you get the internal flights for nothing and the same baggage allowance for your whole journey. LATAM also operates international flights from Europe to Aeroparque (Jorge Newbery), Buenos Aires domestic airport, which means that you can get connecting flights to Patagonia without the headache, and cost, of passing through Ezeiza, the international airport.
As an alternative, especially for non-first time visitors, you can fly to Santiago in Chile and take flights south from the same airport. You can fly to Puerto Montt and cross into Argentina via the Bariloche Lakes Crossing or fly further south to Balmaceda (Coyhaique) and drive in through the amazing Carretera Austral. Cars are usually cheaper to hire in Chile and they are allowed to cross borders, whereas cars rented in Argentina need special paperwork. If you have plenty of time, you can buy a round the world ticket and callin at Easter Island and Tahiti on your way west from South America.
There are somegood deals to be had flying to Uruguay and so you could fly to Montevideo in Uruguay and take the ferry to Buenos Aires. And, finally, for the adventurous and well-heeled, you could take the Falklands Air Bridge from RAF Brize Norton via Ascension Island (or Africa or Cape Verde) and enter Argentina through Rio Gallegos or Punta Arenas. Once you are in Welsh Patagonia, there are no internal flights and, therefore, crossing from the Atlantic to the Andes can only be done by bus or car. In the Andes, Bariloche is served by several airlines from the north and the south, in Esquel, the only option is the (almost) daily flight to Buenos Aires. Trelew has flights to Buenos Aires and to El Calafate and Tierra del Fuego, while Porth Madryn, 65 km to the north, has flights to and from Buenos Aires.