Another person linked to the first Ebola case also tested positive while in quarantine, bringing the total number of infections in Kenya to seven.
The ministry said the four new patients were in isolation and receiving care.
The other two patients were in stable condition.
The first patient died on October 5, days after he arrived in Kenya from the Democratic Republic of Congo, where an outbreak has killed at least 4300 people since May, accounting for nearly half of all confirmed cases there.
The outbreak involves the rare Bundibugyo virus, for which there is no approved vaccine or treatment.
Ebola is highly contagious and spreads through direct contact with the bodily fluids of an infected person, including blood and vomit, or through contact with contaminated materials and surfaces such as bedding and clothing.
About 130 people who had been in contact with the first patient, a 40-year-old Kenyan man, were identified, the ministry said.
Of those, 128 were in quarantine and two were being traced, it said.
Kenya has strengthened screening and established an emergency hotline for anyone who suspects they may have been in contact with an Ebola patient or develops symptoms.
The current Ebola outbreak began in mid-May in eastern Democratic Republic of Congo with health officials confirming on Sunday it had spread to another province.
The National Institute for Biomedical Research confirmed two cases - including one death - in North Ubangi.
It becomes the eighth province affected by the outbreak, alongside North Kivu, Ituri, Haut-Uele, Tshopo, Sud-Kivu, Bas-Uele and South Ubangi.
Congo's latest situation report showed 67 new confirmed cases and 26 new deaths in the 24 hours to October 16, bringing the cumulative total to 9415 cases and 4536 deaths since the outbreak began.
The outbreak remains concentrated in eastern Congo, with Ituri province reporting 44 of the 67 new cases.
with DPA
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