The company, which also owns Bubbaloo and Clorets, is the world's second-biggest chewing gum business after the Wrigley Company.
Cadbury bought Adams from Pfizer, the pharmaceutical giant. However, analysts are suggesting that the figure of $4.2bn was too high for Adams, which had been valued at around $4bn by some industry experts.
Other food giants in the running for the business are thought to have included Nestle and Kraft Foods. However, there were some suggestions that Cadbury had overpaid for the business.
The market for chewing gum is growing faster than chocolate around the world. Even Singapore recently lifted its infamous ban on chewing gum, after pressure from US trade policy negotiators.
Cadbury is a dominant presence in the UK confectionery market, but is relatively small in the US. The company says that the acquisition of Adams will make it joint leader in overall confectionery and number two in chewing gum with a 27% market share. The deal will also give it inroads to the Latin American market.
John Sunderland, CEO of Cadbury Schweppes, said: "Adams gives us confectionery market leadership and a unique portfolio with an offering in every confectionery category. It brings powerful brands, access to new geographies and significant scale in the fastest-growing confectionery sectors. Cost and revenue synergies, and the opportunity to drive the business within a global confectionery group, will create significant value for our share owners."
As well as its famous Dairy Milk chocolate, Cadbury owns brands such as Flake, Crunchie and Picnic.
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