It is a very common condition in contracts that notice etc. must normally terminate at the end of a month or take effect from the beginning of the month. It is usually the same thing.

It probably means midnight, if anyone really wants to make a fuss, but in practice notice usualy ends at the end of working hours on the last working day in the month, and takes effect at the start of working hours on the first working day in the next month. Special rules may apply if there is a weekend or holiday in between.

It saves calculating rent, wages etc. etc. for periods shorter than a month.

Opsigelse ... med en måneds varsel til den første i en måned
A month´s notice to take effect from the first day of a month.
This means that you cannot give less than a calendar month´s notice. If notice is actually announced on any day other than the last day of the month, then in practice more than a month´s notice is given.

It is usually considered bad practice to give notice after working hours on the last day, e.g. by calling or contacting an employee at home.

If notice is actually given, for instance on the 25th of March, then it will take effect on 1 May = the employee is employed until 30th April under notice, and unemployed after that.

An alternate way of expressing more or less the same thing is
X måneders skriftligt varsel til udgangen af en måned.

Here the focus is on the ´varsel´ or period of notice, which ends at the end of the month, but technically the notice takes effect on the first day of the next month.

Jeg er opsagt til den første = I will be unemployed (with effect) from the 1st

Under opsigelse til udgangen af november = working under notice until the end of November.

HTH