We present the results of laser magnetic resonance measurements performed on the ground ^2P state of singly ionized atomic carbon (C_II). The 2^P_(3/2) ← ^2P_(1/2) fine-structure intervals of both ^(12)C^+ and ^(13)C^+ have been determined with a precision of approximately 1 ppm, and the g_J factors to approximately one part in 10^4. Specifically, we find that g_(J=(1/2)) = 0.66576(11) and g_(J=(3/2)) = 1.33412(11), while for ^(12)C^+ ΔE_0(^2P_(3/2) ← ^2P_(1/2))= 1900536.9(1.3) MHz, with ΔE_0(^2P_(3/2) ← ^2P_(1/2)) = 1900545.8(2.1) and ΔE(^2P(3/2) ← ^2P_(1/2), F = 2 ← 1) = 1900466.1(2.3) MHz in ^(13)C^+. The highly precise values of the ^(12)C_II and ^(13)C_II fine-structure intervals verify the already secure far-infrared astronomical identification of C^+ and should allow the interstellar (^(12)C / ^(13)C) ratio to be unambiguously determined in a number of environments.