This Letter presents a search for highly ionizing magnetic monopoles in <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:mn>262</a:mn><a:mtext> </a:mtext><a:mtext> </a:mtext><a:mi mathvariant="normal">μ</a:mi><a:mrow><a:msup><a:mrow><a:mi mathvariant="normal">b</a:mi></a:mrow><a:mrow><a:mo>−</a:mo><a:mn>1</a:mn></a:mrow></a:msup></a:mrow></a:mrow></a:math> of ultraperipheral <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mrow><e:mi>Pb</e:mi><e:mo>+</e:mo><e:mi>Pb</e:mi></e:mrow></e:math> collision data at <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mrow><g:msqrt><g:mrow><g:msub><g:mi>s</g:mi><g:mrow><g:mi>NN</g:mi></g:mrow></g:msub></g:mrow></g:msqrt><g:mo>=</g:mo><g:mn>5.36</g:mn><g:mtext> </g:mtext><g:mtext> </g:mtext><g:mi>TeV</g:mi></g:mrow></g:math> collected by the ATLAS detector at the LHC. A new methodology that exploits the properties of clusters of hits reconstructed in the innermost silicon detector layers is introduced to study highly ionizing particles in heavy-ion data. No significant excess above the background, which is estimated using a data-driven technique, is observed. Using a nonperturbative semiclassical model, upper limits at 95% confidence level are set on the cross section for pair production of monopoles with a single Dirac magnetic charge in the mass range of 20–150 GeV. Depending on the model, monopoles with a single Dirac magnetic charge and mass below 80–120 GeV are excluded. © 2025 CERN, for the ATLAS Collaboration 2025 CERN