The IceCube Neutrino Observatory has been continuously taking data to search for s long neutrino bursts since 2007. Even if a Galactic core-collapse supernova is optically obscured or collapses to a black hole instead of exploding, it will be detectable via the MeV neutrino burst emitted during the collapse. We discuss a search for such events covering the time between 2008 April 17 and 2019 December 31. Considering the average data taking and analysis uptime of 91.7% after all selection cuts, this is equivalent to 10.735 yr of continuous data taking. In order to test the most conservative neutrino production scenario, the selection cuts were optimized for a model based on an 8.8 solar mass progenitor collapsing to an O–Ne–Mg core. Conservative assumptions on the effects of neutrino oscillations in the exploding star were made. The final selection cut was set to ensure that the probability to detect such a supernova within the Milky Way exceeds 99%. No such neutrino burst was found in the data after performing a blind analysis. Hence, a 90% C.L. upper limit on the rate of core-collapse supernovae out to distances of ≈25 kpc was determined to be 0.23 yr⁻¹. For the more distant Magellanic Clouds, only high neutrino luminosity supernovae will be detectable by IceCube, unless external information on the burst time is available. We determined a model-independent limit by parameterizing the dependence on the neutrino luminosity and the energy spectrum.