I will not interpret intercontinental in modern terms but rather view it as trade among distinct civilisations.
Such trade dates back to Ancient Egyptian civilisations, Babylonians and Indus Valley Civilisation.
Proofs exists in form of archeological excavations of, for example , potteries of Indus Civilisation in Iran and other parts of central Asia.
The following articles may be of help: ThinkQuest - Ancient Trade in Mediterranean Wikipedia article(s) on these cultures
The British Museums's website has a very descriptive timeline of ancient trade.
(It is flash
based, cannot directly link.
Navigate your way starting here .)
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There is direct evidence of truly intercontinental trade dating back to approximately 1700 BC.
Jack Turner, in his book Spice: The History of a Temptation , writes that a handful of cloves were found in a charred vessel in Syria.
The find was remarkable in that, until modern times, cloves only grew on five tiny islands that are part of the Moluccas in Indonesia.
Clay tablets located at the same archeological site reference a specific king, allowing the tablets and the cloves to be dated to approximately 1721 BC.
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According to the Wikipedia, Egypt traded with Canaan at the time of the first dynasty and probably even before that, that's about 3000 BCE.
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Obsidian was exported during the Stone Age (Neolithic) from Melos (an island in Europe) to the Near East (Asia) and Egypt (Africa).
Blades, cutting tools, piercing tools, arrow heads, mirrors and artwork are made from obsidian.
The trade started 13000 years ago, before the discovery of agriculture.
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To further detail Lev's example -- there is some evidence of cultural exchange between the Maadi culture of the Nile delta region in Lower Egypt somewhere during the middle of the Naqada period of predynastic Egyptian history, in the neighborhood of 3900-3500 BCE.
Some of the evidence for this in the archaeological record is a type of blade (referred to as the "Canaanite Blade") that shows some Palestinian influences, as well as an apparently simultaneous shift to the usage of certain copper tools rather than their stone counterparts in both this Maadi region as well as in Palestine.
Source: The Oxford History of Ancient Egypt 2003 Edition, pp54-55
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I know this question is old, but the previous answers did not touch on truly intercontinental trade.
The first example of intercontinental trade has to be during the Age of Discovery.
From the 15th-17th century the European nations explored beyond their borders in search of trade, and subsequently subjugation.
The interaction between them and other cultures is the world's first example of intercontinental trade.
The Vikings may have traded with people far beyond their borders, and even the Carthaginians may have reached the New World, but in terms of sustained trade the Age of Discovery has to be paramount.
The exploration missions of the "discoverers" from Europe were all prefaced upon finding a quicker route to the silk trade routes of Asia that Marco Polo originally discovered with his overland route.
This was the impetus for European exploration and would lead to the wide spread European colonization and the development of intercontinental trade in goods previously non-existent in mainland Europe such as spices, and chocolates.
These trade routes, and the driving force behind them have to be the first examples of intercontinental trade because they actually endured and influenced history.
The previous routes were short lived.
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Bronze age:
Troy and Greece: one is in Europe, the other in Asia.
The Greek states did balk at the tolls of Troy, or at least that is one of the reasons given for the war by Herodotus in The Histories .
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Tin from Britain was used in bronze found in the middle east.
And it's likely the Phoenicians were getting copper from across the Atlantic during the same time period.
Humans didn't start using iron for everything because it was better than bronze.
Civilizations collapsed and the trade networks for bronze alloy materials fell apart
so people switched to iron because it was readily available locally.
This was around the "Fall of Troy," invasion of the Sea Peoples, roughly 1200 bc.
Which is the same time that ancient copper mines in Michigan were abandoned.
Humans were originally nomadic so intercontinental trade has been going pretty much as long as we've existed.
It might just stop sometimes because we get stupid every few thousand years.
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Probably this happened several times and maybe even failed and broke up in different regions.  What hints and proofs do we have of the earliest succesful intercontinental trade relations?  I assume there didn't yet exist a common currency system and it was pure barter trade.
First evidence of intercontinental or cross-cultural trade?
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Proof exists in the form of archeological findings that show trade as far back as the 3000 BC, Stone Age, the Age of Discovery and  the Vikings. This is a difficult question to answer since evidence will vary in terms of quality and abundance. However, there is evidence that Egypt traded with other civilisations as early as 3000 BC, and even some stone age acheaological evidence. Many people may point to the Age of Discovery as the most well-documented period of intercontinental trade.
Various archeaological evidence exists pointing to the existence of trade at various points in ancient human history between African and Asian cultures. The Age of Discovery may be the period with the best supported documentation of intercontinental trade. There is direct evidence of intercontinental trade dating back to 1700 BC. Egypt may have traded with Canaan as far back as 3000 BC.