More heavy snow is forecast for the north and northwest overnight with temperatures set to fall as low as -10C.
The national committee monitoring the weather situation has warned that inappropriate driving is now the biggest problem on Ireland's roads.
Inappropriate driving is more dangerous that during heavy snowfalls last week, the Severe Weather Co-ordination Committee said.
It warned ice is now causing a major hazard and people are 'driving way too fast'.
Weather, travel updates, and regional round-up
It also appealed for people to exercise extreme caution on footpaths.
The committee said there has been a huge effort by communities across Ireland to clear footpaths and housing estates and this had led to a dramatic reduction in injuries being reported to hospitals.
The National Roads Authority says salt supplies are being maintained and 28,000 tonnes are still available.
All major transport operators say services are operating with minor disruptions.
Met Éireann says the severe cold temperatures should ease towards the end of this week.
Black ice and some freezing fog are continuing to make driving conditions on motorways, national roads and secondary roads across the country dangerous.
Parts of the west and northwest are particularly badly affected, from Mayo through Sligo to Donegal, and gardaí continue to advise extreme caution on the N17 and N5.
Routes in the northeast are icy, as are the Naas Road and the M50 in Dublin.
In the midlands, road temperatures are staying below -3C in parts of Meath and Westmeath.
Many routes in the southeast remain treacherous, with black ice causing problems on secondary roads in Wicklow, Wexford, Carlow and Kilkenny.
Most Bus Éireann services are running with some delays and diversions, and there are cancellations in Athlone, Dundalk, Drogheda and Mayo.
Last buses from Dublin Bus operated at 9pm both from the city centre and the suburbs.
All Iarnród Éireann routes are running, except for the M3 Parkway.
There is also a full Luas service, and all airports are fully operational.
Despite the slight daytime thaw, footpaths remain icy.
In urban areas with tall buildings, pedestrians in particular are warned to watch out for deposits of compacted snow falling from roofs.
The HSE has said the number of people presenting at A&E Departments with weather-related fractures has reached crisis point.
Separately, people are being urged to give blood to boost supplies during the cold weather.
The Irish Blood Transfusion Service is holding six extra blood donation clinics around the country today - in Cork, Dublin, Dundalk, Limerick and Carlow.
It is hoping to collect about 800 units of blood, in order to avoid any cancellations of elective surgery this week.
