Dr Howard Zucker, acting commissioner of the state Health Department in New York, has wished Dr Spencer a speedy recovery.
Dr Mary Travis Bassett, New York City health commissioner, said Dr Spencer was being monitored in an isolation unit at Bellevue Hospital.
Medical staff at the hospital had been training for the last month on how to deal with Ebola cases, she added.
New York Mayor Bill de Blasio said the city is prepared to tackle Ebola and that proper protocols had been followed with the care of Dr Spencer.
He said Bellevue Hospital in New York is specially designed for the treatment of Ebola patients.
"The patient is in good shape and has gone into a great deal of detail with our personnel as to his actions the last few days so we have a lot to work with," he said.
"We have a patient who has been very communicative and precise and who has only been back a very short time and has been quite clear about individuals he had close contact with."
Brain cancer patient Ashya King has responded well but has not fully recovered as he nears the end of his proton therapy treatment, the clinic treating him said.
The five-year-old was taken to Prague, in the Czech Republic, by his parents to get him the specialist treatment.
The director of the Proton Therapy Center (PTC), Iva Tatounova, told Sky News: "We always thought Ashya would benefit from the specialist treatment here at the PTC.
"And whilst he is still not fully recovered, he's responded very well to the proton beam therapy and there is no reason that he will not continue to get stronger once he leaves here."
Ashya's parents Brett and Naghmeh King sparked an international police hunt when they removed their son from Southampton General Hospital on August 28 without medical consent.
They faced a protracted legal battle to get him to the PTC, with a High Court judge only approving the move after they had been released from police custody in Spain.
Dr Spencer went through enhanced screening when he arrived at JFK airport in New York last week, officials have said.
Special screening was introduced earlier this month at five major US airports including JFK for travelers coming from the three countries in West Africa worst hit by Ebola: Guinea, Liberia and Sierra Leone.
The US Centers for Disease Control said the patient, who had worked with Ebola patients in Guinea, "participated in the enhanced screening for all returning travelers from these countries" on his arrival at JFK.
The doctor "went through multiple layers of screening and did not have a fever or other symptoms of illness," the CDC added in a statement.
The CDC did not name the patient, but he was identified by other officials as Dr. Craig Spencer.
Here are some facts that we know so far about the first person to be diagnosed with Ebola in New York:
- Dr Craig Spencer had recently returned from treating Ebola patients in Guinea, West Africa, with Doctors Without Borders
- The 33-year-old had specialised in international emergency medicine at Columbia University-New York Presbyterian Hospital in New York City since 2011. He had not been to work nor seen any patients since his return
- He left Guinea on October 14, stopped over in Brussels and arrived at John F Kennedy Airport in New York on October 17
- He rode several subways, went bowling and may have eaten at a restaurant before developing symptoms
- He developed symptoms on Wednesday and called authorities on Thursday when they worsened
- Two friends and his fiancee have been quarantined. One of the three is in hospital in isolation. His fiancee is not symptomatic
- His apartment in Harlem is sealed off
- Further tests will be carried out by the Centres for Disease Control and Prevention to confirm the initial results
- His case is the fourth confirmed case in America but the first for New York
A New York doctor who has tested positive for Ebola had close contact with three friends.
Dr Craig Spencer is in isolation at Bellevue Hospital in New York and health officials have sought to reassure the public they are safe from the Ebola virus.
The 33-year-old had recently returned from treating Ebola patients in Guinea, West Africa, with Doctors Without Borders.
A specially trained team wearing protective gear transported the doctor to Bellevue Hospital from his Manhattan apartment on Thursday.
The three friends are now all quarantined, the New York City Health Commissioner said. One of those is also under isolation in hospital.
A fourth contact was a taxi driver, who was not considered at risk, he said.
New York City Mayor Bill de Blasio has confirmed a doctor has tested positive for Ebola, becoming the city's first confirmed case in New York.
He said the doctor had worked with Ebola patients in West Africa.
The Mayor said there was: "No reason for New Yorkers to be alarmed."
He added: "We have been preparing for months for the threat poised by Ebola."
Dr Craig Spencer has tested positive for Ebola, according to reports in the New York Times.
Ebola could kill more than 90,000 people in Liberia by December if there is not a significant increase in international action, a report suggests.
A team of US scientists estimated there could be 90,122 deaths in the country's most populated county Montserrado by December 15.
They added that some 170,996 people may have been infected by the virus by then, representing 12% of an overall population of some 1.38 million.
Lead researcher Professor Alison Galvani, from the School of Public Health at Yale University, said: "Our predictions highlight the rapidly closing window of opportunity for controlling the outbreak and averting a catastrophic toll of new Ebola cases and deaths in the coming months.
"Although we might still be within the midst of what will ultimately be viewed as the early phase of the current outbreak, the possibility of averting calamitous repercussions from an initially delayed and insufficient response is quickly eroding."