Sgt. Conrad was killed and four other commandos were wounded Friday during an operation against the militants, the US military said.
The US military has stepped up its airstrikes in the Horn of Africa nation after getting President Donald Trump's approval for expanded military operations there.
Major Hussein Ali, a Somali military officer told Reuters the attack took place at a military base just outside the town of Kismayu in Southern Somalia.
About 500 USA troops are deployed in Somalia.
The US said its personnel had provided advice, assistance and aerial surveillance during the mission.
The attack, about 50 kilometers north of the port city of Kismayo, came as US forces were helping Somali and Kenyan troops construct the outpost, part of a multiday mission to help clear the area of militants with the al-Shabab terror group, according to USA military officials.
Four other service members were wounded.
Three U.S. service members and the wounded partner force service member were medically evacuated to receive additional treatment, officials said.
The incident is the first known death of a USA servicemen in Africa since the ambush in Niger last October.
Conrad was born in Mesa, Ariz., and joined the Army on June 10, 2010, the statement said. Al-Shabab has been blamed for the truck bombing in Mogadishu that killed more than 500 people last October. The overarching goal in Somalia for the Defense Department is to help the Somali government in providing a safe and secure environment for the Somali population, they added.
The U.S. military and others have expressed concern about the 21,000-strong AU force's plan to withdraw by 2020 and hand over security responsibilities to Somali forces, saying the local troops are not ready.
This is the second US military fatality in Somalia in the a year ago.
Somali officials have said civilians have been killed in more than one joint US military operation with Somali forces.
A U.S. Africa Command statement said the four were in the care of the U.S. Embassy medical team in neighboring Kenya.
A US Defense official confirmed to media that four American soldiers were injured in the attack.