He was not in the office at the time of the stroke.
Deputy Mayor Faye White will assume the role of acting Mayor.
Napier City councillors were on Wednesday united in wishing Dalton a speedy recovery, and sending best wishes to him and his family.
Councillor Annette Brosnan said she had seen Dalton yesterday afternoon and he had appeared well at the time, so was in shock when she heard.
Councillor Tony Jeffery said Napier would be getting behind Dalton in wishing him a quick recovery.
"Bill is very strong and I am sure he will make a full recovery, and we are just praying for that and I am sure the whole city is at the moment."
Other regional leaders who worked closely were also sending their thoughts to Dalton.
Hawke's Bay Regional Council chairman Rex Graham said he and Dalton had been working on their health recently.
The pair used to joke about how well they were each doing on their respective diets, he said.
"He was doing so well," Graham said.
Dalton, who is in his late 60s, is in his second, and final term as Mayor of Napier.
The former stockbroker was elected to the role in 2012, and in the 2016 election ran unopposed.
Dalton has served as a councillor since 2007.
He has presided over changes to the city including the Marine Parade redevelopment and allowing Easter Sunday Trading.
Upcoming projects supported by Dalton include a new pool, and the aquarium expansion.
Recently, he told Hawke's Bay Today he did not have any major plans after he hangs up the mayoral chains in October, but did say he was looking forward to spending more time with his family, especially his grandchildren.
