What can I do to eliminate rediculous outliers in my study's data set?

One of my participants has rediculously long response times. Response time is a dependent measure in my study. With the participant's data (using 1 way ANOVA) I get a p value of .07. Without it I get a p value of .03. So, it's a matter of significant results or not. I only have 27 participants and can't run anymore. Is there a statistical technique to eliminate/reduce the effects of his rediculously long RT? Will this also affect the effect size?