The visa is still valid. In fact, once you get your new passport, the old one will be returned to you as it still bears a valid visa and you will have to travel with both passports.
The US State Department has a great FAQ site which states the following:
If your visa is still valid you can travel to the United States with
your two passports, as long as the visa is valid, not damaged, and is
the appropriate type of visa required for your principal purpose of
travel. (Example: tourist visa, when your principal purpose of travel
is tourism). Both passports (the valid and the expired one with the
visa) should be from the same country and type (Example: both
Uruguayan regular passports, both official passports, etc.). When you
arrive at the U.S. port-of-entry (POE, generally an airport or land
border) the Customs and Border Protection Immigration Officer will
check your visa in the old passport and if s/he decides to admit you
into the United States they will stamp your new passport with an
admission stamp along with the annotation "VIOPP" (visa in other
passport). Do not try to remove the visa from your old passport and
stick it into the new valid passport. If you do so, your visa will no
longer be valid.
For your name change, it is recommended to get a new visa with your new name, but your old visa remains valid.
If your name has legally changed through marriage, divorce, or a court
ordered name change, you will need to obtain a new passport. Once you
have a new passport, the Department of State recommends that you apply
for a new U.S. visa to make it easier for you to travel to and from
the United States.
You should travel with evidence of your name change (such as a court document - make sure it is in English) in case you are asked.
If you require to have your name changed on your visa, just go through the process of applying for a non-immigrant visa. Once you reach the interview, explain your circumstances. Make sure you bring evidence of your name change.