Ten Mice, a Thousand Flasks

A lab has $1000$ flasks of serum and knows exactly one is contaminated. The contaminant is tasteless and shows its effect in exactly $24$ hours — no sooner, no later. All sampling must be done right now, and there is exactly one chance to read results: tomorrow. Mice can each sip from as many flasks as needed. What is the minimum number of mice that guarantees pinpointing the contaminated flask, and how must they drink?