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.
For the latest rugby union news follow @bbcrugbyunion on Twitter.

From 20 March, speed restrictions and speed cameras will be introduced where work is taking place.
The upgrade work will see the six-mile "missing link" completed on the M8 in North Lanarkshire.
Improvements to the Raith Interchange and widening of key sections of the M8, M73 and M74 are also taking place.
The work will see upgrading of the M8 between Baillieston and Newhouse to motorway standard.
This will require upgrades to the connections to the M73 motorway at Baillieston and a new junction at Shawhead, which will connect to the A725 East Kilbride road.
There will also be new junctions at Eurocentral and Chapelhall.
Improvements to the Raith Interchange near Hamilton in Lanarkshire include realignment of the A725.
There will be an underpass for the A725 and bridges carrying the Raith roundabout.
The improvements are expected to cut journey times by up to 18 minutes for the busiest sections of the M8.
Graeme Reid, project sponsor for Transport Scotland, said: "We have reached a pivotal point in this three year project, involving traffic management measures which are likely to cause some disruption to road users, especially at peak times.
"The biggest challenge on a project of this scale is to keep the 100,000 vehicles that use these key routes every day moving and to ensure disruption to road users is minimised as far as is practically possible."
He added: "The installation of safety cameras is a tried and tested mechanism that improves safety for both road users and the workforce during what will be an extremely busy period in terms of construction."
The first traffic restrictions and safety cameras will be brought in along the M74 northbound and southbound from J6 Hamilton to J3A Daldowie.
Three narrow lanes will be in operation to allow the construction of a new lane and hard shoulder.
Improvements at the Raith Interchange will see speed restrictions and temporary lane closures and diversions to allow excavation works to start on the construction of the A725 underpass below the existing roundabout.
The new underpass will link the Bellshill Bypass and the East Kilbride Expressway (A725) and create a new route for traffic, cutting out the heavily-congested Raith Junction.
Two lanes will remain operational in each direction across all roads during peak times, although traffic management and speed restrictions will be in place.
London Welsh have avoided liquidation after paying their debts, and are now set to be taken over by an United States-based investment group.
[NEXT_CONCEPT]
Traffic restrictions are being brought in next week on some of Scotland's busiest motorways as work ramps-up on a £500m improvement scheme.