The Senate passed the bill late last week.
Governor Rick Perry initially declined the money because it would require changes to the state's unemployment system in order to be eligible for funds.
The Senate has approved a series of reforms to the system to receive the federal money for unemployment insurance.
The measure would change a person's unemployment benefit from one that calculates payments based on the four fiscal quarters previous to the most recent one, to a calculation based on the most recent four quarters.
It would also extend benefits to part-time workers who lose their jobs and to those who quit their jobs based on compelling family reasons.
Perry has said the changes would increase the tax burden on employers.
The Texas Public Policy Foundation earlier praised Perry's announcement declining the additional stimulus funds for unemployment insurance.
The Foundation's Center for Fiscal Policy Director Talmadge Heflin said legislators should keep in mind that every additional dollar Texas employers have to pay for people who aren't working is one less dollar available for job creation and economic recovery.
The state's non-agricultural employment fell by 47,000 in March, following a loss of 52,400 jobs in February.