No. Thermodynamic functions of the phase transitions of water explain the science behind this question.
Long term, a definite No. The question does have a valid common sense reference answer: Short term, Yes. But only for a short time. Also its not a perfect sphere and most wouldn't refer to it as a bubble. Think of simple ice cube tray in the freezer that is removed before fully frozen with some liquid still in the center. Temperature at the core hasn't reached freezing point yet. Technically a pocket of liquid water and air within a solid mass of water.
Thinking also in the other direction, the same could be said of a water vapor bubble momentarily existing in a container of boiling water. Long term, a definite No.
Using a water phase diagram chart used to show the physical states of water at different temperatures and pressures, the phase boundaries reveal the singular conditions of temperature and pressure where liquid water, gaseous water and hexagonal ice stably coexist, there is a 'triple point' if you look closely. (chart can be found here)> http://www.lsbu.ac.uk/water/phase.html However it is impossible to create a water bubble within an ice bubble for any long term time period.