The investment giant has agreed a deal to control the £267 million Income Focus fund, two months after it was frozen by administrator Link Fund Solutions.
Link had been considering options for the fund since October, when Woodford decided to close his business and resign from managing the fund.
The administrator told investors on Friday that it decided ASI, the largest active fund manager in the UK, is best placed to add value. It said the asset manager has provided commitments that it intends to grow the assets in the fund and attract additional investment.
Management of the fund will switch over to ASI at the end of December, although the fund will remain suspended until February. It will be renamed as the ASI Income Focus fund and will be managed by Thomas Moore and Charles Luke from the firm.
The income fund was the last ran by the high-profile stock picker, whose Patient Capital fund was recently taken over by Schroders.
Andrew Millington, head of UK equities at ASI, said: "We recognise the issues that may have weighed on the minds of investors in the fund over the past months and we are resolute in our determination to turn things around for them.
"We will reposition the fund into a focused portfolio of our favoured, liquid UK equity income names, aiming to generate sustainable long-term income growth for its investors.
"The scale of our UK equity business means we will manage turnover of stocks to minimise costs and set the fund on the right track for 2020 and beyond."