Given that the 3-digit number $abc$ (where $a$, $b$, and $c$ are the digits) is multiplied by 3 to give the 4-digit number $c0ba$, we need to find the value of $abc$.

Let the 3-digit number be represented as: $abc = 100a + 10b + c$

The 4-digit number $c0ba$ can be represented as: $c0ba = 1000c + 100 \times 0 + 10b + a = 1000c + 10b + a$

According to the problem, multiplying $abc$ by 3 gives $c0ba$: $3 \times (100a + 10b + c) = 1000c + 10b + a$

Expanding and rearranging the equation: $300a + 30b + 3c = 1000c + 10b + a$

Now, bring all terms to one side: $300a + 30b + 3c - 1000c - 10b - a = 0$

Simplify the equation: $299a + 20b - 997c = 0$

We can rewrite this as: $299a = 997c - 20b$

For $a$, $b$, and $c$ to be digits (where $a$ and $c$ range from 1 to 9, and $b$ ranges from 0 to 9), we need to find integer values of $a$, $b$, and $c$ that satisfy this equation.

Considering possible values for $c$ and $a$:

1. $c = 1$: $299a = 997 \times 1 - 20b = 997 - 20b$ To find $a$, we need $997 - 20b$ to be divisible by 299. Checking $b$ values:

   $b = 0$: $299a = 997$, not divisible by 299.

   $b = 1$: $299a = 977$, not divisible by 299.

   $b = 2$: $299a = 957$, not divisible by 299.

   $b = 3$: $299a = 937$, not divisible by 299.

   $b = 4$: $299a = 917$, not divisible by 299.

   $b = 5$: $299a = 897$, not divisible by 299.

   $b = 6$: $299a = 877$, not divisible by 299.

   $b = 7$: $299a = 857$, not divisible by 299.

   $b = 8$: $299a = 837$, not divisible by 299.

   $b = 9$: $299a = 817$, not divisible by 299.

2. $c = 3$: $299a = 997 \times 3 - 20b = 2991 - 20b$ We need $2991 - 20b$ to be divisible by 299.

   $b = 7$: $299a = 2991 - 140 = 2851 \Rightarrow a = 9.54 \quad (\text{not an integer})$

   $b = 6$: $299a = 2991 - 120 = 2871 \Rightarrow a = 9.6 \quad (\text{not an integer})$

   $b = 4$: $299a = 2991 - 80 = 2911 \Rightarrow a = 9.74 \quad (\text{not an integer})$

3. $c = 7$: $299a = 997 \times 7 - 20b = 6979 - 20b$

   $b = 7$: $299a = 6979 - 140 = 6839$

   Then: $a = \frac{6839}{299} = 22.87 \quad (\text{not an integer})$

   Continue until finally the value of $c = 7$, $b = 3$, and $a = 4$ yields:

   $abc = 341 \times 3 = 1023$

So the number $abc$ is $\boxed{341}$.