DAY local controller described ground conflict event during very low visibility when landing traffic runway exit clearance resulted in an opposite direction traffic situation with an aircraft taxiing for departure.

Narrative

I was working Local/Ground Control combined; which is a normal configuration that we work in the Tower. I don't know how long I was on position but it was less than an hour. Air Carrier X checked on with me on final for 6L and was given the wind; RVR; and cleared to land. Air Carrier Y called for taxi and was given taxi instructions to 6L via Taxiway U and R and to report when he was turning on to Taxiway R. The weather at the airport was VV002 and 1/2 visibility which are the minimums for CAT 1 approaches for the runway and all RVR's were around 1200-1400. When Air Carrier X landed I was unable to see where he and Air Carrier Y were at and could only hear Air Carrier X 's reverse thrusters noise so I figured that he was between Taxiway T and U so I issued taxi instructions for him to exit the runway at Taxiway M and proceed to the ramp from Taxiway M. Air Carrier X then reported that he was exiting at U and was now nose to nose with Air Carrier Y. Air Carrier X said that he was going to hold short of Taxiway R so I instructed Air Carrier X to turn left on Taxiway R and then turn on M to the ramp. Air Carrier Y reported that Air Carrier X was clear and continued his taxi. When Air Carrier X got to his gate he reported that when he was on short final he saw Air Carrier Y on his TCAS and it appeared that he was close to the runway. I don't feel that this was any ones fault but with the visibility being so low; I thought it needed to be addressed. Recommendation; DAY is building a new Tower with the height of it being around 237ft. The current one we are in is less than 100ft. The visibility is going to be a big factor in the safety of ground movements when the ceilings/visibility get that low and it does fairly often. I would suggest that some type of ground RADAR be looked into or installed before an actual incident/accident happens.