Dr Mike Cruz, the government official leading the coronavirus response, said the woman was a relative of a person who recently returned to the territory with Covid-19.
He said the woman had "multiple co-morbidities", including end-stage renal disease, which compromised her immune system.
Dr Cruz said there are 15 confirmed cases of the virus on Guam, with the government on Sunday morning declaring a major disaster.
"Five of the first 14 confirmed cases had recent travel, most of which is linked to the Philippines," Dr Cruz told a news conference on Sunday afternoon.
"There is strong evidence that Covid-19 cases has spread throughout our community and has affected residents in the northern, central and southern villages," he added.
With 15 cases, Guam is the country or territory in the Pacific with the most number of people affected by Covid-19.
On Sunday, the governor, Lou Leon Guerrero, asked for a major disaster to be declared in the US territory, which would open the way for significant federal funding.
Ms Leon Guerrero said in a statement that she had asked president Donald Trump for stimulus to help a tourism-based economy in freefall.
Already, most of Guam is in a state of lockdown, with schools and all public facilities closed since the middle of last week.
At the news conference alongside Dr Cruz, Ms Leon Guerrero had a message for Guamanians: "Nothing is much more important than you staying home. Please stay home, let's ride this together."