It is well known (see Dvoretzky, Erdos and Kakutani (Bull. Res. Council Israel Sect. F 7F (1958) 175–180) and Le Gall (J. Funct. Anal. 71 (1987) 246–262)) that a planar Brownian motion $(B_{t})_{t\ge 0}$ has points of infinite multiplicity, and these points form a dense set on the range. Our main result is the construction of a family of random measures, denoted by $\{{\mathcal{M}}_{\infty }^{\alpha }\}_{0<\alpha <2}$, that are supported by the set of the points of infinite multiplicity. We prove that for any $\alpha \in (0,2)$, almost surely the Hausdorff dimension of ${\mathcal{M}}_{\infty }^{\alpha }$ equals $2-\alpha $, and ${\mathcal{M}}_{\infty }^{\alpha }$ is supported by the set of thick points defined in Bass, Burdzy and Khoshnevisan (Ann. Probab. 22 (1994) 566–625) as well as by that defined in Dembo, Peres, Rosen and Zeitouni (Acta Math. 186 (2001) 239–270).