The original location of Delaware's city hall is unknown. The first records of established government offices date back to 1868 where the police station was established for $3.50 per month in a room over J.L. Latimer's music store. Additional records from January 1876, show that the second floor over the City Drug Store, at the corner of William St. and Sandusky St., was rented for use as a Council Room for $50 per year.
On May 8, 1879, the City Council met and proposed to purchase property on the SE corner of William and Sandusky Streets to build a city hall. The City Council felt that $10,000 would be sufficient money to construct the building. Over the next few months blueprints were reviewed and eventually they voted accepted Mr. F.F. Snitzer's (or Switzer, or Schnitzer, depending on what information you're reading) plans, and began construction. The original estimate of $10,000 was only enough to build the first floor of the new building, and on December 1, 1879, construction was stopped. The City Council asked for an additional $40,000 on the spring ballot, which greatly upset the citizens of Delaware. More cost overruns and delays ensued, and from the time of its completion in 1880 until its untimely demise in 1934, the city hall was known as "The White Elephant."
After the city hall was built, it quickly fell into disrepair, and became almost unusable. The citizens of Delaware wanted action to be taken, since they were already unhappy with the situation surrounding its construction. On Feb 24, 1934, there we no more issues concerning the city hall and what to do with it, as it burned to the ground under suspicious circumstances. No definite cause was found for the fire, but it is believed that it may have been the act of disgruntled townspeople.
Delaware's current city hall was built after the fire, at the same location as the old one. The corner stone from the original building escaped the fire, and people urged the city to incorporate it into the new building, but it was unceremoniously dumped in the Olentangy River. The new building was completed 3 years to the day after the fire. It is a smaller, more efficient building, and also built to be more fire resistant as well. In 1992 the city hall underwent extensive renovations. At that time the cornerstone from the original building was incorporated into the structure.
Pictures from the city hall fire can be found here. Also a more in depth description of the original city hall, along with newspaper articles can be found at delawareohio.us.