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If you live in the DUblin area, you need to send in your completed application form to the Driver Licensing Section at Dublin City Council (Chancery Street, Dublin 1). To my knowledge, they will not re-issue a new driving licence to you on the spot - given the volume of applciations, they normally take about 10 working days to process a new licence.
If you live outside of the Dublin area, bring your completed application form to the Motor Tax office of your local authority. They may be able to re-issue you with a new licence faster than this.
Since your motor insurance will expire at the end of this week - a third option is to contact your Motor Tax office and tell them you have lost your licence. They may agree to issue a letter to state you hold a full current driving licence for insurance purposes.
Other than that, I don't think you will have any problems once you have applied for your new licence. I have never been requested to provide evidence on an annual basis to my motor insurer that I hold a full driving licence. (Then again, I have been insured with the same insurance company for a number of years).
Hope this helps