The agreed offer was unveiled by Electronics Boutique chief executive John Steinbrecher, who said he had wanted a deal for three years.
Electronics Boutique has 182 stores compared to Game's 86 outlets, with the move creating a retail leader in the fast growing craze for video games.
Under the planned takeover, unveiled on Monday, the Game brand would be developed separately from its new owners.
Mr Steinbrecher said: "The games market continues to grow strongly. Already this year we have seen the fastest selling PlayStation and PC entertainment titles ever."
Darker stores
With the growth in demand for video games expected to continue he added that there would be no redundancies or shop closures because Game already ran a "tight ship".
The two chains also catered for different customers with Game's darker stores and large video screens targeting a younger crowd than Electronics Boutique whose bright and open stores aimed to attract a wider range of general High Street shoppers.
The company said it planned to use the acquisition to act as a springboard into continental Europe.
The offer, accepted by Game's directors, values the target's shares at 134.3p, a premium of 31% to Game's value on the stock market on Friday.
Rising profits
While higher than Game's previous share price the offer is still well below the value of the company when it floated on the stock market last year at 200p.
Game's share price fell heavily after it warned in January that it had suffered from a Christmas shortage of Sony PlayStation and other games consoles.
The offer came as Electronics Boutique announced its final results for the year to 31 January which showed a 92% rise in pre-tax profit to £15.3m from £8m.
Like-for-like sales were up 10.7% during the year with the company saying that the first six weeks of the current year had started well with sales up 10% on the same basis.
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