To solve the optimization problem described, we first need to convert the natural language description into a symbolic representation. This involves defining variables and formulating the objective function and constraints using these variables.

Let's define:
- $x_1$ as the number of full-time employees.
- $x_2$ as the number of part-time employees.

The objective is to minimize the weekly wage bill, which can be represented as $800x_1 + 400x_2$ since full-time employees earn $800 per week and part-time employees earn $400 per week.

The constraints are:
1. The total number of employees should be at least 60: $x_1 + x_2 \geq 60$.
2. At least 40 employees must be full-time: $x_1 \geq 40$.
3. The number of full-time employees should be at least a third of the number of part-time employees: $x_1 \geq \frac{1}{3}x_2$.
4. The weekly wage bill should be below $30,000: $800x_1 + 400x_2 \leq 30000$.

In symbolic representation, we have:
```json
{
  'sym_variables': [('x1', 'number of full-time employees'), ('x2', 'number of part-time employees')],
  'objective_function': 'minimize 800*x1 + 400*x2',
  'constraints': [
    'x1 + x2 >= 60',
    'x1 >= 40',
    'x1 >= (1/3)*x2',
    '800*x1 + 400*x2 <= 30000'
  ]
}
```

Now, let's implement this problem using Gurobi in Python:

```python
from gurobipy import *

# Create a new model
m = Model("Iota_Software")

# Define variables
x1 = m.addVar(vtype=GRB.INTEGER, name="full_time_employees")
x2 = m.addVar(vtype=GRB.INTEGER, name="part_time_employees")

# Set the objective function
m.setObjective(800*x1 + 400*x2, GRB.MINIMIZE)

# Add constraints
m.addConstr(x1 + x2 >= 60, "total_employees")
m.addConstr(x1 >= 40, "full_time_min")
m.addConstr(x1 >= (1/3)*x2, "full_time_vs_part_time")
m.addConstr(800*x1 + 400*x2 <= 30000, "wage_bill")

# Optimize the model
m.optimize()

# Print the solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Full-time employees: {x1.x}")
    print(f"Part-time employees: {x2.x}")
    print(f"Wage bill: ${800*x1.x + 400*x2.x}")
else:
    print("No optimal solution found.")
```