We report the first measurements of the anisotropic upper critical field $H_{c2}(T)$ for K$_{2}$Cr$_{3}$As$_{3}$ single crystals up to 60 T and $T > 0.6$ K. Our results show that the upper critical field parallel to the Cr chains, $H_{c2}^parallel (T)$, exhibits a paramagnetically-limited behavior, whereas the shape of the $H_{c2}^perp (T)$ curve (perpendicular to the Cr chains) has no evidence of paramagnetic effects. As a result, the curves $H_{c2}^perp (T)$ and $H_{c2}^parallel(T)$ cross at $Tapprox 4$ K, so that the anisotropy parameter $gamma_H(T)=H_{c2}^perp/H_{c2}^parallel (T)$ increases from $gamma_H(T_c)approx 0.35$ near $T_c$ to $gamma_H(0)approx 1.7$ at 0.6 K. This behavior of $H_{c2}^|(T)$ is inconsistent with triplet superconductivity but suggests a form of singlet superconductivity with the electron spins locked onto the direction of Cr chains.
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