This resulted in the unauthorised access to the details of 45 of the 145,000 user accounts on the database, including names, dates of birth and telephone numbers.
The City Council is working with the external company which hosts and provides the database and the Information Commissioner's Office to investigate the incident.
Fiona Brown, Executive Director of Neighbourhoods, said: "Following confirmation of the incident by the system's suppliers, we are working with them to thoroughly investigate what happened and implement any remedial measures necessary to minimise the chances of this happening again.
"While there is no evidence to suggest that this data has been used inappropriately, we are urging library users to be extra vigilant especially when putting any personal details online or responding to any requests for their information if they're unsure who the request has come from.
"At this stage of the investigation the external company which provides and hosts the database hasn't been able to identify which 45 customers' details have been accessed, which is why we are urging all library users to be vigilant.
"The type of data accessed can be used as a starting point to seek further information about a person to commit identity theft or fraud.
"We take the issue of cyber security extremely seriously and have robust procedures in place to guard against data loss including the use of encryption and other practical measures.
"That includes reviewing and reinforcing existing security measures, and liaising with the Information Commissioner's Office and other regulatory authorities.
"Cyber Crime is a constantly evolving and fast-moving arena. In common with all organisations, large and small, we face daily challenges to adapt to the pace, volume and sophistication of attacks.
"But we have extensive technological and organisational measures to secure our customers' personal information, and require all our contractors to provide evidence that they have the same measures in place."
Additional advice and guidance is available on the Council website:
Members of the public who require further information, can contact the Council's Data Protection Office on 0191 561 1023 or via email@example.com