Of the 34 new cases, 15 are from Nagaon, six from Golaghat, three each from Barpeta, Baksa and Sonitpur and one each from Chirang, West Karbi Anglong, Dibrugarh and Lakhimpur.
Two persons have died of the disease this week, raising the death toll to six in the state, the minister said.
A 67-year-old cancer patient, who recently returned from Mumbai, died due to COVID-19 at Tezpur Medical College and Hospital on Wednesday, and another Chennai-returnee succumbed to the disease on Monday.
Of the total 3,319 cases, 2,061 are active patients, 1,249 have recovered, six have died and three migrated out of the state.
As many as 152 more patients were discharged on Wednesday, while the highest single-day recovery was recorded on Tuesday, with 313 patients reported cured of COVID-19.
Assam has witnessed a surge in COVID-19 cases since inter-state movement of people was allowed on May 4, with more than 3,000 cases being registered since then. Prior to that, it had just 42 coronavirus cases.
The state breached the 1000-mark in coronavirus cases on May 29, and the 3000-mark in just about 10 days, on June 9.
Assam currently has 86 containment zones, with Cachar having the maximum number of such areas at 33, followed by Kamrup (Metro) at 19, Golaghat at six, Dhubri and Darrang five each, Goalpara, Kamrup (Rural) and Majuli three each, Karbi Anglong two, and Charaideo, Karimganj, Kokrajhar, Hojai, Nagaon, Baksa and Bongaigaon one each.
As many as 1,69,842 samples have been examined till date, of which 1,59,7600 have tested negative, the state Health and Family Welfare Department said in its bulletin.
Meanwhile, keeping in view the rise in COVID-19 cases, the health department said it would conduct the second round of Assam Community Surveillance Programme (ACSP), covering all villages and wards of the state.
The health workers will visit people's homes to identify flu-like cases, respiratory infections like Severe Acute Respiratory Infections (SARI), influenza-like illness (ILI) and vector-borne diseases such as Japanese Encephalitis, malaria and dengue.
The first round was held last month and 29,771 villages were covered during the session, with only 26,000 people showing signs of fever, but no case of COVID-19 was reported, the minister had earlier said. DG RMS RHL