A winding-up petition was dismissed by the High Court on Monday.
The brief hearing in London revealed an undisclosed amount of money owed to Her Majesty's Revenue and Customs had been paid by the Exiles.
A club statement said the takeover had been agreed, subject to the approval of the Rugby Football Union.
The Championship side but have struggled to pay players' wages since previously avoiding liquidation in April, and Rowland Phillips has not been replaced since leaving as head coach.
Welsh - one of England's oldest clubs, having been formed in 1885 - said the financial security provided by the unnamed California-based group would allow the club "to plan for the future with confidence and renewed enthusiasm".
Club chairman Bleddyn Phillips commented: "We are thrilled and very excited to be able to announce the arrival of a new owner which will provide the club with the financial strength to sustain its ambition to be not only a competitive rugby team playing in the top tiers of the game in the United Kingdom, but, equally importantly, to consolidate and enhance its social, community and amateur based activities at its spiritual home at Old Deer Park".
The Exiles were a Premiership side as recently as 2015, when they were based at the Kassam Stadium in Oxford and finished bottom of the table after losing all 22 games.
They subsequently returned to their traditional home at Old Deer Park in Richmond after three seasons away and started the new season with a narrow 33-32 defeat at Rotherham on Sunday.
Their plight further highlights the difficult financial situation in English rugby's second tier.
The Rugby Football Union, which runs the Championship, has been accused of under-funding the league, which was reduced from 16 teams to 12 in 2009.
Every team became fully professional at that time, and a new five-year funding deal with the governing body was agreed in 2015.
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London Welsh have avoided liquidation after paying their debts, and are now set to be taken over by an United States-based investment group.