We construct and study properties of an infinite dimensional analog of Kahane's theory of Gaussian multiplicative chaos \cite{K85}. Namely, if $H_T(ω)$ is a random field defined w.r.t. space-time white noise $\dot B$ and integrated w.r.t. Brownian paths in $d\geq 3$, we consider the renormalized exponential, weighted w.r.t. the Wiener measure $\mathbb P_0$. We construct the almost sure limit $μ_γ$ in the {\it entire weak disorder (subcritical)} regime and call it {\it subcritical GMC} on the Wiener space. We show that $$ μ_γ\Big\{ω: \lim_{T\to\infty} \frac{H_T(ω)}{T(φ\starφ)(0)} \ne γ\Big\}=0 \qquad \mbox{almost surely,} $$ meaning, $μ_γ$ is supported only on $γ$-{\it thick paths}, and consequently, the normalized version is singular w.r.t. the Wiener measure. We characterize uniquely the limit $μ_γ$ w.r.t. the mollification scheme $φ$ in the sense of Shamov \cite{S14} and the random {\it rooted} measure $\mathbb Q_{μ_γ}(d\dot B dω)= μ_γ(dω,\dot B)P(d\dot B)$. We then determine the fractal properties of the measure around $γ$-thick paths: $-C_2 \leq \liminf_{r\to 0} r^2 \log \widehatμ_γ(\|ω\| < r) \leq \limsup_{r\to 0}\sup_ηr^2 \log \widehatμ_γ(\|ω-η\| < r) \leq -C_1$ w.r.t a weighted norm $\|\cdot\|$. Here $C_1>0$ and $C_2<\infty$ are the uniform upper (resp. pointwise lower) Hölder exponents which are {\it explicit} in the entire weak disorder regime. Moreover, they converge to the scaling exponent of the Wiener measure as the disorder approaches zero. Finally, we establish negative and $L^p$ ($p>1$) moments for the total mass of $μ_γ$ in the weak disorder regime.